| L(s) = 1 | + 2·2-s − 6·3-s + 3·4-s − 12·6-s + 26·7-s − 6·8-s + 21·9-s + 4·11-s − 18·12-s + 52·14-s − 12·16-s − 36·17-s + 42·18-s + 30·19-s − 156·21-s + 8·22-s − 40·23-s + 36·24-s − 54·27-s + 78·28-s − 8·29-s − 42·32-s − 24·33-s − 72·34-s + 63·36-s − 50·37-s + 60·38-s + ⋯ |
| L(s) = 1 | + 2-s − 2·3-s + 3/4·4-s − 2·6-s + 26/7·7-s − 3/4·8-s + 7/3·9-s + 4/11·11-s − 3/2·12-s + 26/7·14-s − 3/4·16-s − 2.11·17-s + 7/3·18-s + 1.57·19-s − 7.42·21-s + 4/11·22-s − 1.73·23-s + 3/2·24-s − 2·27-s + 2.78·28-s − 0.275·29-s − 1.31·32-s − 0.727·33-s − 2.11·34-s + 7/4·36-s − 1.35·37-s + 1.57·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.845930012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.845930012\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T + T^{2} + 5 p T^{3} - 23 T^{4} + 5 p^{3} T^{5} + p^{4} T^{6} - p^{7} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T - 206 T^{2} + 80 T^{3} + 32707 T^{4} + 80 p^{2} T^{5} - 206 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 382 T^{2} + 72003 T^{4} - 382 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 36 T + 1046 T^{2} + 22104 T^{3} + 418323 T^{4} + 22104 p^{2} T^{5} + 1046 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 30 T + 449 T^{2} - 4470 T^{3} + 180 T^{4} - 4470 p^{2} T^{5} + 449 p^{4} T^{6} - 30 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 20 T - 129 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 822 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 + 770 T^{2} - 330621 T^{4} + 770 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 50 T - 263 T^{2} + 1250 T^{3} + 2337508 T^{4} + 1250 p^{2} T^{5} - 263 p^{4} T^{6} + 50 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 932 T^{2} + 1635078 T^{4} + 932 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 100 T + 6102 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 144 T + 12410 T^{2} - 791712 T^{3} + 40616931 T^{4} - 791712 p^{2} T^{5} + 12410 p^{4} T^{6} - 144 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 44 T + 1234 T^{2} + 216304 T^{3} - 12835901 T^{4} + 216304 p^{2} T^{5} + 1234 p^{4} T^{6} - 44 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 24 T + 1370 T^{2} + 28272 T^{3} - 10061325 T^{4} + 28272 p^{2} T^{5} + 1370 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 234 T + 29609 T^{2} + 2657538 T^{3} + 183051300 T^{4} + 2657538 p^{2} T^{5} + 29609 p^{4} T^{6} + 234 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 2 T - 311 T^{2} - 17326 T^{3} - 20069852 T^{4} - 17326 p^{2} T^{5} - 311 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 52 T + 8358 T^{2} + 52 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 90 T + 13745 T^{2} - 994050 T^{3} + 107982084 T^{4} - 994050 p^{2} T^{5} + 13745 p^{4} T^{6} - 90 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 38 T - 9863 T^{2} + 44650 T^{3} + 79886164 T^{4} + 44650 p^{2} T^{5} - 9863 p^{4} T^{6} - 38 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 20500 T^{2} + 187537542 T^{4} - 20500 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 72 T + 16202 T^{2} + 1042128 T^{3} + 160441923 T^{4} + 1042128 p^{2} T^{5} + 16202 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 15166 T^{2} + 198604227 T^{4} - 15166 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−7.48648631182100294558106381320, −7.47714660131268725102083522968, −7.21196020824932607001040810802, −6.63144955488260325116389368751, −6.53002075501113332818534225022, −6.50484221001224225464639250015, −5.88625612494286435374801272161, −5.63575755339058352695631919696, −5.60909758371248619408964207249, −5.57458008704128769616970542677, −5.14478201505659318763634002656, −4.76631040383731409946451209957, −4.67045841760122013299132733327, −4.52237972443779369621808524106, −4.28138655270854586376728366600, −4.03026283542381001134254350325, −3.39079579403300998126143041405, −3.31337112171294096463724065765, −2.91342508322134079630602762481, −1.99549176162360804685915768498, −1.97868151263975733608444574943, −1.76987334655824090688519683427, −1.59671249791477935225535253902, −0.62373707054128203683210487197, −0.53477630185195833696425007371,
0.53477630185195833696425007371, 0.62373707054128203683210487197, 1.59671249791477935225535253902, 1.76987334655824090688519683427, 1.97868151263975733608444574943, 1.99549176162360804685915768498, 2.91342508322134079630602762481, 3.31337112171294096463724065765, 3.39079579403300998126143041405, 4.03026283542381001134254350325, 4.28138655270854586376728366600, 4.52237972443779369621808524106, 4.67045841760122013299132733327, 4.76631040383731409946451209957, 5.14478201505659318763634002656, 5.57458008704128769616970542677, 5.60909758371248619408964207249, 5.63575755339058352695631919696, 5.88625612494286435374801272161, 6.50484221001224225464639250015, 6.53002075501113332818534225022, 6.63144955488260325116389368751, 7.21196020824932607001040810802, 7.47714660131268725102083522968, 7.48648631182100294558106381320
