| L(s) = 1 | + 8·2-s + 2·3-s + 48·4-s + 16·6-s − 156·7-s + 256·8-s − 320·9-s − 512·11-s + 96·12-s + 1.28e3·13-s − 1.24e3·14-s + 1.28e3·16-s − 1.08e3·17-s − 2.56e3·18-s + 722·19-s − 312·21-s − 4.09e3·22-s + 2.22e3·23-s + 512·24-s + 1.02e4·26-s − 802·27-s − 7.48e3·28-s − 2.56e3·29-s − 1.90e3·31-s + 6.14e3·32-s − 1.02e3·33-s − 8.64e3·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.128·3-s + 3/2·4-s + 0.181·6-s − 1.20·7-s + 1.41·8-s − 1.31·9-s − 1.27·11-s + 0.192·12-s + 2.11·13-s − 1.70·14-s + 5/4·16-s − 0.906·17-s − 1.86·18-s + 0.458·19-s − 0.154·21-s − 1.80·22-s + 0.876·23-s + 0.181·24-s + 2.98·26-s − 0.211·27-s − 1.80·28-s − 0.567·29-s − 0.356·31-s + 1.06·32-s − 0.163·33-s − 1.28·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 4 p^{4} T^{2} - 2 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 156 T + 37090 T^{2} + 156 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 512 T + 308746 T^{2} + 512 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1286 T + 1154568 T^{2} - 1286 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1080 T + 27142 T^{2} + 1080 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2224 T + 8710018 T^{2} - 2224 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2568 T + 42618142 T^{2} + 2568 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 1908 T + 1202506 p T^{2} + 1908 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 7806 T + 153893776 T^{2} + 7806 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3060 T + 175373302 T^{2} - 3060 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2596 T + 294439418 T^{2} - 2596 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14412 T + 476722882 T^{2} - 14412 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9638 T + 778138360 T^{2} - 9638 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 43008 T + 1863517414 T^{2} + 43008 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 25852 T + 1080210866 T^{2} + 25852 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 27466 T + 2743897916 T^{2} - 27466 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 87780 T + 4786920070 T^{2} + 87780 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 25904 T + 3322669062 T^{2} - 25904 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3720 T + 6120640510 T^{2} + 3720 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 38320 T + 5670980698 T^{2} + 38320 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 167264 T + 15588224134 T^{2} + 167264 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4038 T + 16759910608 T^{2} - 4038 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−8.889950874212584637566860976004, −8.812441517575173384549810804771, −8.161125888643428871837873866703, −7.87427289729321177756250479883, −7.08420223237480691098586413505, −6.94619378034008457914273692127, −6.29192329275537246990688909344, −5.99869319411129929653605789764, −5.57590717559780903453889600319, −5.40208248985451997518783818322, −4.65199309325899105457027284888, −4.18088820898706767240663947713, −3.53178685107702324869245905818, −3.29323357258722488846242837747, −2.81507395889817277689515608763, −2.53824981914850203473990909202, −1.66835304052240589824172915064, −1.11907159253655898791215686190, 0, 0,
1.11907159253655898791215686190, 1.66835304052240589824172915064, 2.53824981914850203473990909202, 2.81507395889817277689515608763, 3.29323357258722488846242837747, 3.53178685107702324869245905818, 4.18088820898706767240663947713, 4.65199309325899105457027284888, 5.40208248985451997518783818322, 5.57590717559780903453889600319, 5.99869319411129929653605789764, 6.29192329275537246990688909344, 6.94619378034008457914273692127, 7.08420223237480691098586413505, 7.87427289729321177756250479883, 8.161125888643428871837873866703, 8.812441517575173384549810804771, 8.889950874212584637566860976004
