| L(s) = 1 | − 3-s − 2·4-s − 5-s − 4.12·7-s + 9-s − 0.438·11-s + 2·12-s − 13-s + 15-s + 4·16-s + 0.438·17-s − 2·19-s + 2·20-s + 4.12·21-s − 0.438·23-s + 25-s − 27-s + 8.24·28-s + 1.43·29-s − 31-s + 0.438·33-s + 4.12·35-s − 2·36-s + 2.68·37-s + 39-s − 5·41-s − 0.561·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s − 1.55·7-s + 0.333·9-s − 0.132·11-s + 0.577·12-s − 0.277·13-s + 0.258·15-s + 16-s + 0.106·17-s − 0.458·19-s + 0.447·20-s + 0.899·21-s − 0.0914·23-s + 0.200·25-s − 0.192·27-s + 1.55·28-s + 0.267·29-s − 0.179·31-s + 0.0763·33-s + 0.696·35-s − 0.333·36-s + 0.441·37-s + 0.160·39-s − 0.780·41-s − 0.0856·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 7 | \( 1 + 4.12T + 7T^{2} \) |
| 11 | \( 1 + 0.438T + 11T^{2} \) |
| 17 | \( 1 - 0.438T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 0.438T + 23T^{2} \) |
| 29 | \( 1 - 1.43T + 29T^{2} \) |
| 37 | \( 1 - 2.68T + 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 + 0.561T + 43T^{2} \) |
| 47 | \( 1 - 2.87T + 47T^{2} \) |
| 53 | \( 1 - 5.56T + 53T^{2} \) |
| 59 | \( 1 - 7.68T + 59T^{2} \) |
| 61 | \( 1 + 2.43T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 0.684T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 5.80T + 89T^{2} \) |
| 97 | \( 1 - 4.12T + 97T^{2} \) |
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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
−7.75262721920477284861265522841, −6.85720809615408402036589089989, −6.35388204070931616797382114283, −5.50628971539588182115868012494, −4.88560392158614023197511025602, −3.93701417366125744202168332049, −3.52161426361953551491311278479, −2.45334781522915018782560218585, −0.836961252723423981718720555260, 0,
0.836961252723423981718720555260, 2.45334781522915018782560218585, 3.52161426361953551491311278479, 3.93701417366125744202168332049, 4.88560392158614023197511025602, 5.50628971539588182115868012494, 6.35388204070931616797382114283, 6.85720809615408402036589089989, 7.75262721920477284861265522841
