| L(s) = 1 | − 64·2-s − 1.02e3·3-s + 4.32e3·5-s + 6.56e4·6-s + 2.62e5·8-s + 1.59e6·9-s − 2.76e5·10-s + 8.78e6·11-s + 4.08e7·13-s − 4.43e6·15-s − 1.67e7·16-s + 1.71e6·17-s − 1.02e8·18-s − 1.09e8·19-s − 5.62e8·22-s + 6.46e8·23-s − 2.68e8·24-s + 1.22e9·25-s − 2.61e9·26-s − 3.82e9·27-s + 1.45e9·29-s + 2.83e8·30-s + 1.02e9·31-s − 9.01e9·33-s − 1.10e8·34-s − 1.42e10·37-s + 7.02e9·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.812·3-s + 0.123·5-s + 0.574·6-s + 0.353·8-s + 9-s − 0.0874·10-s + 1.49·11-s + 2.34·13-s − 0.100·15-s − 1/4·16-s + 0.0172·17-s − 0.707·18-s − 0.534·19-s − 1.05·22-s + 0.910·23-s − 0.287·24-s + 25-s − 1.65·26-s − 1.90·27-s + 0.455·29-s + 0.0710·30-s + 0.208·31-s − 1.21·33-s − 0.0122·34-s − 0.911·37-s + 0.378·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(2.010931981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.010931981\) |
| \(L(\frac{15}{2})\) |
|
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 | $C_2$ | \( 1 + p^{6} T + p^{12} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 38 p^{3} T - 743 p^{6} T^{2} + 38 p^{16} T^{3} + p^{26} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 864 p T - 48081629 p^{2} T^{2} - 864 p^{14} T^{3} + p^{26} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 8787312 T + 42694140041413 T^{2} - 8787312 p^{13} T^{3} + p^{26} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 20420932 T + p^{13} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 1719462 T - 9901621483336493 T^{2} - 1719462 p^{13} T^{3} + p^{26} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 109702942 T - 30018247978801695 T^{2} + 109702942 p^{13} T^{3} + p^{26} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 646760160 T - 85737657373241783 T^{2} - 646760160 p^{13} T^{3} + p^{26} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 728867274 T + p^{13} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1028049116 T - 23360661312536661135 T^{2} - 1028049116 p^{13} T^{3} + p^{26} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 14229390962 T - 41093657066634019953 T^{2} + 14229390962 p^{13} T^{3} + p^{26} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 44544458406 T + p^{13} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 54689828968 T + p^{13} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 47868325716 T - \)\(31\!\cdots\!71\)\( T^{2} - 47868325716 p^{13} T^{3} + p^{26} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 169986882858 T + \)\(28\!\cdots\!91\)\( T^{2} - 169986882858 p^{13} T^{3} + p^{26} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 300765540198 T - \)\(14\!\cdots\!75\)\( T^{2} + 300765540198 p^{13} T^{3} + p^{26} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 369996272360 T - \)\(25\!\cdots\!81\)\( T^{2} - 369996272360 p^{13} T^{3} + p^{26} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 787010801908 T + \)\(71\!\cdots\!77\)\( T^{2} - 787010801908 p^{13} T^{3} + p^{26} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 559441472256 T + p^{13} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 121137579650 T - \)\(16\!\cdots\!33\)\( T^{2} - 121137579650 p^{13} T^{3} + p^{26} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 290426785064 T - \)\(45\!\cdots\!43\)\( T^{2} + 290426785064 p^{13} T^{3} + p^{26} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 3965105603046 T + p^{13} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6025919250630 T + \)\(14\!\cdots\!31\)\( T^{2} + 6025919250630 p^{13} T^{3} + p^{26} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 11302818199190 T + p^{13} T^{2} )^{2} \) |
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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
−11.71214800219690408380829736369, −11.04347210025940194188550659564, −10.38741913924268999053597143186, −10.31227065174134196231315541456, −9.289033583800896213528401013880, −9.100972738243814031760981522617, −8.276091784516939875750192602495, −8.253195853930942092376106261060, −6.88511939718976379848072479054, −6.64357774131558038604454100453, −6.50799051055870502888176635863, −5.40751852382742602106715316276, −5.10275559847559588667890028124, −4.10126480337009116614034836887, −3.76605930041484420798162292126, −3.18790515163779347031753690166, −1.73001184347645447569747961200, −1.58946161854468050858559789255, −0.995448120222954595862324476649, −0.44108690700528761403665989144,
0.44108690700528761403665989144, 0.995448120222954595862324476649, 1.58946161854468050858559789255, 1.73001184347645447569747961200, 3.18790515163779347031753690166, 3.76605930041484420798162292126, 4.10126480337009116614034836887, 5.10275559847559588667890028124, 5.40751852382742602106715316276, 6.50799051055870502888176635863, 6.64357774131558038604454100453, 6.88511939718976379848072479054, 8.253195853930942092376106261060, 8.276091784516939875750192602495, 9.100972738243814031760981522617, 9.289033583800896213528401013880, 10.31227065174134196231315541456, 10.38741913924268999053597143186, 11.04347210025940194188550659564, 11.71214800219690408380829736369
