| L(s) = 1 | − 1.80·2-s + 3-s + 1.25·4-s + 5-s − 1.80·6-s − 3.38·7-s + 1.33·8-s + 9-s − 1.80·10-s − 2.61·11-s + 1.25·12-s + 1.48·13-s + 6.11·14-s + 15-s − 4.93·16-s − 6.99·17-s − 1.80·18-s + 0.688·19-s + 1.25·20-s − 3.38·21-s + 4.72·22-s + 1.33·24-s + 25-s − 2.67·26-s + 27-s − 4.26·28-s − 5.00·29-s + ⋯ |
| L(s) = 1 | − 1.27·2-s + 0.577·3-s + 0.629·4-s + 0.447·5-s − 0.736·6-s − 1.28·7-s + 0.473·8-s + 0.333·9-s − 0.570·10-s − 0.789·11-s + 0.363·12-s + 0.411·13-s + 1.63·14-s + 0.258·15-s − 1.23·16-s − 1.69·17-s − 0.425·18-s + 0.157·19-s + 0.281·20-s − 0.739·21-s + 1.00·22-s + 0.273·24-s + 0.200·25-s − 0.525·26-s + 0.192·27-s − 0.806·28-s − 0.929·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 1.80T + 2T^{2} \) |
| 7 | \( 1 + 3.38T + 7T^{2} \) |
| 11 | \( 1 + 2.61T + 11T^{2} \) |
| 13 | \( 1 - 1.48T + 13T^{2} \) |
| 17 | \( 1 + 6.99T + 17T^{2} \) |
| 19 | \( 1 - 0.688T + 19T^{2} \) |
| 29 | \( 1 + 5.00T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 - 3.16T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 1.03T + 43T^{2} \) |
| 47 | \( 1 - 8.10T + 47T^{2} \) |
| 53 | \( 1 - 13.7T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 5.43T + 67T^{2} \) |
| 71 | \( 1 - 0.00276T + 71T^{2} \) |
| 73 | \( 1 + 0.808T + 73T^{2} \) |
| 79 | \( 1 + 9.68T + 79T^{2} \) |
| 83 | \( 1 + 9.82T + 83T^{2} \) |
| 89 | \( 1 + 0.0682T + 89T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
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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
−7.58317885877114317131499028175, −7.03182083171904478239708452450, −6.36509916691953025444785091486, −5.63054801894586533823091608602, −4.50475995890284376991376487930, −3.85597117049928210310690839053, −2.62837808188845742092701881273, −2.33782390491893938530924318321, −1.04990307309377534819509352657, 0,
1.04990307309377534819509352657, 2.33782390491893938530924318321, 2.62837808188845742092701881273, 3.85597117049928210310690839053, 4.50475995890284376991376487930, 5.63054801894586533823091608602, 6.36509916691953025444785091486, 7.03182083171904478239708452450, 7.58317885877114317131499028175
