| L(s) = 1 | − 1.23e3·3-s + 5.74e4·5-s + 6.42e4·7-s − 6.66e4·9-s − 2.46e6·11-s − 8.03e6·13-s − 7.10e7·15-s + 7.11e7·17-s − 1.36e8·19-s − 7.93e7·21-s − 1.18e9·23-s + 2.07e9·25-s + 2.05e9·27-s + 8.90e8·29-s + 4.59e9·31-s + 3.04e9·33-s + 3.69e9·35-s + 1.95e10·37-s + 9.92e9·39-s − 2.72e9·41-s − 5.17e10·43-s − 3.82e9·45-s − 5.35e10·47-s − 9.27e10·49-s − 8.78e10·51-s − 8.26e10·53-s − 1.41e11·55-s + ⋯ |
| L(s) = 1 | − 0.978·3-s + 1.64·5-s + 0.206·7-s − 0.0417·9-s − 0.419·11-s − 0.461·13-s − 1.60·15-s + 0.714·17-s − 0.664·19-s − 0.201·21-s − 1.67·23-s + 1.70·25-s + 1.01·27-s + 0.278·29-s + 0.930·31-s + 0.410·33-s + 0.339·35-s + 1.25·37-s + 0.451·39-s − 0.0895·41-s − 1.24·43-s − 0.0687·45-s − 0.724·47-s − 0.957·49-s − 0.699·51-s − 0.512·53-s − 0.689·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 412 p T + p^{13} T^{2} \) |
| 5 | \( 1 - 2298 p^{2} T + p^{13} T^{2} \) |
| 7 | \( 1 - 9176 p T + p^{13} T^{2} \) |
| 11 | \( 1 + 224052 p T + p^{13} T^{2} \) |
| 13 | \( 1 + 8032766 T + p^{13} T^{2} \) |
| 17 | \( 1 - 71112402 T + p^{13} T^{2} \) |
| 19 | \( 1 + 136337060 T + p^{13} T^{2} \) |
| 23 | \( 1 + 1186563144 T + p^{13} T^{2} \) |
| 29 | \( 1 - 890583090 T + p^{13} T^{2} \) |
| 31 | \( 1 - 4595552672 T + p^{13} T^{2} \) |
| 37 | \( 1 - 19585053898 T + p^{13} T^{2} \) |
| 41 | \( 1 + 2724170358 T + p^{13} T^{2} \) |
| 43 | \( 1 + 51762321116 T + p^{13} T^{2} \) |
| 47 | \( 1 + 53572833168 T + p^{13} T^{2} \) |
| 53 | \( 1 + 82633440006 T + p^{13} T^{2} \) |
| 59 | \( 1 - 394266352980 T + p^{13} T^{2} \) |
| 61 | \( 1 + 671061772142 T + p^{13} T^{2} \) |
| 67 | \( 1 + 388156449812 T + p^{13} T^{2} \) |
| 71 | \( 1 + 388772243928 T + p^{13} T^{2} \) |
| 73 | \( 1 - 1540972938026 T + p^{13} T^{2} \) |
| 79 | \( 1 + 3306509559280 T + p^{13} T^{2} \) |
| 83 | \( 1 + 4931756967396 T + p^{13} T^{2} \) |
| 89 | \( 1 - 3502949738490 T + p^{13} T^{2} \) |
| 97 | \( 1 + 388932598558 T + p^{13} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.63835246385214559442450158986, −10.37273141796770652051572379143, −9.760619644000914083258276677262, −8.199141252053687123525657064659, −6.44339503175077823930351706326, −5.76619620170711758643872438644, −4.78201702262356014215964655302, −2.64589147043702641986688409984, −1.46839551040305908006019464381, 0,
1.46839551040305908006019464381, 2.64589147043702641986688409984, 4.78201702262356014215964655302, 5.76619620170711758643872438644, 6.44339503175077823930351706326, 8.199141252053687123525657064659, 9.760619644000914083258276677262, 10.37273141796770652051572379143, 11.63835246385214559442450158986
