| L(s) = 1 | − 1.59·2-s − 3-s + 0.529·4-s + 3.24·5-s + 1.59·6-s − 7-s + 2.33·8-s + 9-s − 5.16·10-s + 1.62·11-s − 0.529·12-s − 0.895·13-s + 1.59·14-s − 3.24·15-s − 4.77·16-s − 1.17·17-s − 1.59·18-s − 3.87·19-s + 1.71·20-s + 21-s − 2.58·22-s − 0.189·23-s − 2.33·24-s + 5.55·25-s + 1.42·26-s − 27-s − 0.529·28-s + ⋯ |
| L(s) = 1 | − 1.12·2-s − 0.577·3-s + 0.264·4-s + 1.45·5-s + 0.649·6-s − 0.377·7-s + 0.827·8-s + 0.333·9-s − 1.63·10-s + 0.489·11-s − 0.152·12-s − 0.248·13-s + 0.425·14-s − 0.838·15-s − 1.19·16-s − 0.285·17-s − 0.374·18-s − 0.888·19-s + 0.384·20-s + 0.218·21-s − 0.550·22-s − 0.0396·23-s − 0.477·24-s + 1.11·25-s + 0.279·26-s − 0.192·27-s − 0.0999·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 1.59T + 2T^{2} \) |
| 5 | \( 1 - 3.24T + 5T^{2} \) |
| 11 | \( 1 - 1.62T + 11T^{2} \) |
| 13 | \( 1 + 0.895T + 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 19 | \( 1 + 3.87T + 19T^{2} \) |
| 23 | \( 1 + 0.189T + 23T^{2} \) |
| 29 | \( 1 + 5.15T + 29T^{2} \) |
| 31 | \( 1 - 5.55T + 31T^{2} \) |
| 37 | \( 1 - 3.72T + 37T^{2} \) |
| 41 | \( 1 + 5.04T + 41T^{2} \) |
| 43 | \( 1 - 4.17T + 43T^{2} \) |
| 47 | \( 1 - 7.08T + 47T^{2} \) |
| 53 | \( 1 + 3.53T + 53T^{2} \) |
| 59 | \( 1 - 2.34T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 15.0T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 1.36T + 83T^{2} \) |
| 89 | \( 1 + 18.4T + 89T^{2} \) |
| 97 | \( 1 - 4.60T + 97T^{2} \) |
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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
−7.37225752771343732241048742975, −6.90577966741045606246688182410, −6.08010200308501042536813568808, −5.71108926159773194760487196884, −4.69124710100266776619761963534, −4.10048077235529637421242991247, −2.71154738074190729124803835648, −1.90130985746293628358311062318, −1.16971023162291648823537322157, 0,
1.16971023162291648823537322157, 1.90130985746293628358311062318, 2.71154738074190729124803835648, 4.10048077235529637421242991247, 4.69124710100266776619761963534, 5.71108926159773194760487196884, 6.08010200308501042536813568808, 6.90577966741045606246688182410, 7.37225752771343732241048742975
