| L(s) = 1 | − 1.45·2-s − 3-s + 0.114·4-s − 5-s + 1.45·6-s + 3.05·7-s + 2.74·8-s + 9-s + 1.45·10-s − 0.438·11-s − 0.114·12-s − 13-s − 4.44·14-s + 15-s − 4.21·16-s − 2.30·17-s − 1.45·18-s + 1.98·19-s − 0.114·20-s − 3.05·21-s + 0.638·22-s + 6.52·23-s − 2.74·24-s + 25-s + 1.45·26-s − 27-s + 0.349·28-s + ⋯ |
| L(s) = 1 | − 1.02·2-s − 0.577·3-s + 0.0570·4-s − 0.447·5-s + 0.593·6-s + 1.15·7-s + 0.969·8-s + 0.333·9-s + 0.459·10-s − 0.132·11-s − 0.0329·12-s − 0.277·13-s − 1.18·14-s + 0.258·15-s − 1.05·16-s − 0.558·17-s − 0.342·18-s + 0.454·19-s − 0.0255·20-s − 0.667·21-s + 0.136·22-s + 1.35·23-s − 0.559·24-s + 0.200·25-s + 0.285·26-s − 0.192·27-s + 0.0659·28-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 | \( 1 + 1.45T + 2T^{2} \) |
| 7 | \( 1 - 3.05T + 7T^{2} \) |
| 11 | \( 1 + 0.438T + 11T^{2} \) |
| 17 | \( 1 + 2.30T + 17T^{2} \) |
| 19 | \( 1 - 1.98T + 19T^{2} \) |
| 23 | \( 1 - 6.52T + 23T^{2} \) |
| 29 | \( 1 - 1.09T + 29T^{2} \) |
| 37 | \( 1 + 3.49T + 37T^{2} \) |
| 41 | \( 1 - 0.573T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 8.21T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 1.03T + 59T^{2} \) |
| 61 | \( 1 + 1.03T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 2.55T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 1.52T + 79T^{2} \) |
| 83 | \( 1 - 4.80T + 83T^{2} \) |
| 89 | \( 1 + 7.11T + 89T^{2} \) |
| 97 | \( 1 - 3.24T + 97T^{2} \) |
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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
−7.69191342951101814425382020007, −7.32651234214709195758778300974, −6.56003889408155243910699828436, −5.39869340522295589362238087118, −4.81227102184382581899604952638, −4.35040577300963793152765407649, −3.15099510429629625850595783073, −1.86888732721672148424891998264, −1.11437528578424438241641825007, 0,
1.11437528578424438241641825007, 1.86888732721672148424891998264, 3.15099510429629625850595783073, 4.35040577300963793152765407649, 4.81227102184382581899604952638, 5.39869340522295589362238087118, 6.56003889408155243910699828436, 7.32651234214709195758778300974, 7.69191342951101814425382020007
