| L(s) = 1 | − 14·3-s + 42·5-s + 894·9-s − 7.42e3·11-s − 2.36e4·13-s − 588·15-s + 1.57e4·17-s + 2.66e4·19-s − 3.26e4·23-s + 1.24e5·25-s + 2.64e4·27-s − 3.16e5·29-s − 1.80e5·31-s + 1.03e5·33-s + 4.58e4·37-s + 3.31e5·39-s + 6.43e5·41-s + 2.04e6·43-s + 3.75e4·45-s + 1.66e6·47-s − 2.21e5·51-s + 4.10e5·53-s − 3.11e5·55-s − 3.72e5·57-s + 1.70e6·59-s − 5.47e5·61-s − 9.93e5·65-s + ⋯ |
| L(s) = 1 | − 0.299·3-s + 0.150·5-s + 0.408·9-s − 1.68·11-s − 2.98·13-s − 0.0449·15-s + 0.779·17-s + 0.890·19-s − 0.559·23-s + 1.59·25-s + 0.258·27-s − 2.40·29-s − 1.08·31-s + 0.503·33-s + 0.148·37-s + 0.894·39-s + 1.45·41-s + 3.92·43-s + 0.0614·45-s + 2.34·47-s − 0.233·51-s + 0.378·53-s − 0.252·55-s − 0.266·57-s + 1.07·59-s − 0.308·61-s − 0.448·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.903552649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.903552649\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 14 T - 698 T^{2} - 16240 p T^{3} - 490265 p^{2} T^{4} - 16240 p^{8} T^{5} - 698 p^{14} T^{6} + 14 p^{21} T^{7} + p^{28} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 42 T - 123166 T^{2} + 263088 p T^{3} + 374647071 p^{2} T^{4} + 263088 p^{8} T^{5} - 123166 p^{14} T^{6} - 42 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 7428 T + 8632202 T^{2} + 56219857920 T^{3} + 711291094304619 T^{4} + 56219857920 p^{7} T^{5} + 8632202 p^{14} T^{6} + 7428 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 70 p^{2} T + 160198410 T^{2} + 70 p^{9} T^{3} + p^{14} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 15792 T - 16280606 p T^{2} + 4651056365760 T^{3} + 6130718726782755 T^{4} + 4651056365760 p^{7} T^{5} - 16280606 p^{15} T^{6} - 15792 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 26614 T - 1253942090 T^{2} - 4644239023312 T^{3} + 2418273122215699231 T^{4} - 4644239023312 p^{7} T^{5} - 1253942090 p^{14} T^{6} - 26614 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 32640 T + 2057178482 T^{2} - 254639647088640 T^{3} - 14236548990664871085 T^{4} - 254639647088640 p^{7} T^{5} + 2057178482 p^{14} T^{6} + 32640 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 158016 T + 39988772806 T^{2} + 158016 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 180740 T - 5817373358 T^{2} - 2989603578895360 T^{3} - \)\(17\!\cdots\!57\)\( T^{4} - 2989603578895360 p^{7} T^{5} - 5817373358 p^{14} T^{6} + 180740 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 45824 T + 20185374250 T^{2} + 9529068243880960 T^{3} - \)\(88\!\cdots\!21\)\( T^{4} + 9529068243880960 p^{7} T^{5} + 20185374250 p^{14} T^{6} - 45824 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 321720 T + 181439440606 T^{2} - 321720 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 1023868 T + 671194246566 T^{2} - 1023868 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1665972 T + 1099984870706 T^{2} - 1103259291706623744 T^{3} + \)\(11\!\cdots\!23\)\( T^{4} - 1103259291706623744 p^{7} T^{5} + 1099984870706 p^{14} T^{6} - 1665972 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 410628 T - 2165589726190 T^{2} + 6248608032034800 T^{3} + \)\(38\!\cdots\!99\)\( T^{4} + 6248608032034800 p^{7} T^{5} - 2165589726190 p^{14} T^{6} - 410628 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1702134 T - 1128927975562 T^{2} + 1618924907272816080 T^{3} + \)\(28\!\cdots\!99\)\( T^{4} + 1618924907272816080 p^{7} T^{5} - 1128927975562 p^{14} T^{6} - 1702134 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 547526 T - 5309538579326 T^{2} - 370216478913573040 T^{3} + \)\(20\!\cdots\!67\)\( T^{4} - 370216478913573040 p^{7} T^{5} - 5309538579326 p^{14} T^{6} + 547526 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2590616 T + 2056450789210 T^{2} + 19343048712604086400 T^{3} - \)\(55\!\cdots\!01\)\( T^{4} + 19343048712604086400 p^{7} T^{5} + 2056450789210 p^{14} T^{6} - 2590616 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4129272 T + 22218672158062 T^{2} - 4129272 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 8008868 T + 26118740970730 T^{2} + 1747516196782370000 p T^{3} + \)\(11\!\cdots\!31\)\( p^{2} T^{4} + 1747516196782370000 p^{8} T^{5} + 26118740970730 p^{14} T^{6} + 8008868 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 2470456 T + 19757905667170 T^{2} - \)\(12\!\cdots\!12\)\( T^{3} - \)\(29\!\cdots\!49\)\( T^{4} - \)\(12\!\cdots\!12\)\( p^{7} T^{5} + 19757905667170 p^{14} T^{6} + 2470456 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 9900786 T + 68835957963214 T^{2} - 9900786 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 15423492 T + 94566779700986 T^{2} - \)\(84\!\cdots\!40\)\( T^{3} + \)\(80\!\cdots\!87\)\( T^{4} - \)\(84\!\cdots\!40\)\( p^{7} T^{5} + 94566779700986 p^{14} T^{6} - 15423492 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 17377472 T + 164552259333822 T^{2} - 17377472 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.70104550776557944944898344050, −7.40037343241422406961754195505, −7.30902306289940021185878171534, −7.28985246771308818698677641825, −7.03503218981262130220260253290, −6.18874456559915966245969273110, −6.10365673643291093943405340781, −5.78943584648158753039347791396, −5.57364347261514177000946544966, −5.28528784141347941295761509725, −4.87053457832152162721405372955, −4.75400070048858678017954387518, −4.72504937983569954058748158577, −4.00333840680333740177532797361, −3.73243684450409317502394283715, −3.43063295030872095826499338234, −3.07598752514072215909753051672, −2.46253907578939283267205469826, −2.25810018513896790548440165901, −2.21423970157092315197588024821, −2.11830791649777830270543068506, −0.977529502770832894696557062749, −0.842447520547978161789360708025, −0.68671340316128552120876657741, −0.24913077485483568455998933811,
0.24913077485483568455998933811, 0.68671340316128552120876657741, 0.842447520547978161789360708025, 0.977529502770832894696557062749, 2.11830791649777830270543068506, 2.21423970157092315197588024821, 2.25810018513896790548440165901, 2.46253907578939283267205469826, 3.07598752514072215909753051672, 3.43063295030872095826499338234, 3.73243684450409317502394283715, 4.00333840680333740177532797361, 4.72504937983569954058748158577, 4.75400070048858678017954387518, 4.87053457832152162721405372955, 5.28528784141347941295761509725, 5.57364347261514177000946544966, 5.78943584648158753039347791396, 6.10365673643291093943405340781, 6.18874456559915966245969273110, 7.03503218981262130220260253290, 7.28985246771308818698677641825, 7.30902306289940021185878171534, 7.40037343241422406961754195505, 7.70104550776557944944898344050
