| L(s) = 1 | − 264·5-s + 686·7-s + 4.98e3·11-s − 1.01e4·13-s − 1.78e4·17-s + 6.25e3·19-s − 1.40e4·23-s + 2.73e4·25-s − 2.43e5·29-s + 4.70e5·31-s − 1.81e5·35-s + 3.11e5·37-s − 9.19e5·41-s − 1.12e5·43-s + 1.02e5·47-s + 3.52e5·49-s − 2.72e6·53-s − 1.31e6·55-s + 1.90e4·59-s − 9.25e5·61-s + 2.67e6·65-s − 2.05e6·67-s + 8.69e5·71-s + 3.50e6·73-s + 3.41e6·77-s − 6.64e6·79-s − 7.85e6·83-s + ⋯ |
| L(s) = 1 | − 0.944·5-s + 0.755·7-s + 1.12·11-s − 1.28·13-s − 0.880·17-s + 0.209·19-s − 0.240·23-s + 0.350·25-s − 1.85·29-s + 2.83·31-s − 0.713·35-s + 1.00·37-s − 2.08·41-s − 0.216·43-s + 0.143·47-s + 3/7·49-s − 2.50·53-s − 1.06·55-s + 0.0120·59-s − 0.521·61-s + 1.21·65-s − 0.834·67-s + 0.288·71-s + 1.05·73-s + 0.852·77-s − 1.51·79-s − 1.50·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 264 T + 8462 p T^{2} + 264 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4980 T + 38737606 T^{2} - 4980 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 10148 T + 75576846 T^{2} + 10148 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 17832 T + 345159502 T^{2} + 17832 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6256 T + 1754965926 T^{2} - 6256 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 14052 T + 6233591566 T^{2} + 14052 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 243588 T + 49005121054 T^{2} + 243588 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 470824 T + 110401476030 T^{2} - 470824 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 311116 T + 134140188030 T^{2} - 311116 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 919248 T + 579002453902 T^{2} + 919248 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 112616 T + 296638211478 T^{2} + 112616 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 102456 T + 463813338910 T^{2} - 102456 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2720028 T + 3726830950846 T^{2} + 2720028 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 19008 T - 1663332768746 T^{2} - 19008 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 925148 T + 2505443574174 T^{2} + 925148 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2053424 T + 3053533142214 T^{2} + 2053424 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 869508 T + 14567098422382 T^{2} - 869508 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3505228 T + 11289516695814 T^{2} - 3505228 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6640856 T + 37612376152158 T^{2} + 6640856 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 7856760 T + 50010766932550 T^{2} + 7856760 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9330384 T + 109971781045486 T^{2} - 9330384 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2220212 T + 72228119801046 T^{2} + 2220212 p^{7} T^{3} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.57526851572705075479432081697, −10.09044807573988639224395868718, −9.497938331525630254815875111195, −9.202171084692448943749368721434, −8.495096268585817997387706021945, −8.071878988251383391394120417095, −7.67721206045429886701212209214, −7.19742349634444734554011499595, −6.42100940558754194034163469971, −6.35661771896507738405290366689, −5.14765097337015763652225361744, −4.93954190247693560846597331913, −4.17244927374621037037280666814, −3.99813506580270940881136839651, −3.03444279862126322902583779027, −2.51020183650228963671319030700, −1.61919959239364366267299229392, −1.19770680018989848264435622719, 0, 0,
1.19770680018989848264435622719, 1.61919959239364366267299229392, 2.51020183650228963671319030700, 3.03444279862126322902583779027, 3.99813506580270940881136839651, 4.17244927374621037037280666814, 4.93954190247693560846597331913, 5.14765097337015763652225361744, 6.35661771896507738405290366689, 6.42100940558754194034163469971, 7.19742349634444734554011499595, 7.67721206045429886701212209214, 8.071878988251383391394120417095, 8.495096268585817997387706021945, 9.202171084692448943749368721434, 9.497938331525630254815875111195, 10.09044807573988639224395868718, 10.57526851572705075479432081697
