| L(s) = 1 | − 2.69·2-s + 5.28·4-s − 5-s + 0.342·7-s − 8.86·8-s + 2.69·10-s − 11-s + 4.11·13-s − 0.924·14-s + 13.3·16-s + 2.11·17-s + 19-s − 5.28·20-s + 2.69·22-s + 3.85·23-s + 25-s − 11.0·26-s + 1.81·28-s + 2.82·29-s + 0.00585·31-s − 18.2·32-s − 5.70·34-s − 0.342·35-s − 1.43·37-s − 2.69·38-s + 8.86·40-s − 5.67·41-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 2.64·4-s − 0.447·5-s + 0.129·7-s − 3.13·8-s + 0.853·10-s − 0.301·11-s + 1.14·13-s − 0.247·14-s + 3.33·16-s + 0.512·17-s + 0.229·19-s − 1.18·20-s + 0.575·22-s + 0.804·23-s + 0.200·25-s − 2.17·26-s + 0.342·28-s + 0.524·29-s + 0.00105·31-s − 3.23·32-s − 0.977·34-s − 0.0579·35-s − 0.235·37-s − 0.437·38-s + 1.40·40-s − 0.887·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 7 | \( 1 - 0.342T + 7T^{2} \) |
| 13 | \( 1 - 4.11T + 13T^{2} \) |
| 17 | \( 1 - 2.11T + 17T^{2} \) |
| 23 | \( 1 - 3.85T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 - 0.00585T + 31T^{2} \) |
| 37 | \( 1 + 1.43T + 37T^{2} \) |
| 41 | \( 1 + 5.67T + 41T^{2} \) |
| 43 | \( 1 - 1.32T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 + 2.23T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 4.61T + 71T^{2} \) |
| 73 | \( 1 + 7.55T + 73T^{2} \) |
| 79 | \( 1 + 9.55T + 79T^{2} \) |
| 83 | \( 1 + 8.96T + 83T^{2} \) |
| 89 | \( 1 + 9.38T + 89T^{2} \) |
| 97 | \( 1 - 3.61T + 97T^{2} \) |
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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
−7.52927844058164484063091373369, −7.01311649702563871679273679094, −6.29075033515083300070443456416, −5.64116402760730277235455510885, −4.59544424949330384644202885986, −3.35411630972299533415719326599, −2.92954210416026498001458745506, −1.70830792752253897748248929774, −1.10479400707550860082188805988, 0,
1.10479400707550860082188805988, 1.70830792752253897748248929774, 2.92954210416026498001458745506, 3.35411630972299533415719326599, 4.59544424949330384644202885986, 5.64116402760730277235455510885, 6.29075033515083300070443456416, 7.01311649702563871679273679094, 7.52927844058164484063091373369
