| L(s) = 1 | − 2-s + 2.16·3-s + 4-s + 4.16·5-s − 2.16·6-s − 0.683·7-s − 8-s + 1.68·9-s − 4.16·10-s − 11-s + 2.16·12-s − 2.68·13-s + 0.683·14-s + 9.01·15-s + 16-s + 3.36·17-s − 1.68·18-s + 4.16·20-s − 1.48·21-s + 22-s − 1.36·23-s − 2.16·24-s + 12.3·25-s + 2.68·26-s − 2.84·27-s − 0.683·28-s − 2.79·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.24·3-s + 0.5·4-s + 1.86·5-s − 0.883·6-s − 0.258·7-s − 0.353·8-s + 0.561·9-s − 1.31·10-s − 0.301·11-s + 0.624·12-s − 0.744·13-s + 0.182·14-s + 2.32·15-s + 0.250·16-s + 0.816·17-s − 0.396·18-s + 0.931·20-s − 0.323·21-s + 0.213·22-s − 0.285·23-s − 0.441·24-s + 2.46·25-s + 0.526·26-s − 0.548·27-s − 0.129·28-s − 0.519·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.330308103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.330308103\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2.16T + 3T^{2} \) |
| 5 | \( 1 - 4.16T + 5T^{2} \) |
| 7 | \( 1 + 0.683T + 7T^{2} \) |
| 13 | \( 1 + 2.68T + 13T^{2} \) |
| 17 | \( 1 - 3.36T + 17T^{2} \) |
| 23 | \( 1 + 1.36T + 23T^{2} \) |
| 29 | \( 1 + 2.79T + 29T^{2} \) |
| 31 | \( 1 + 5.01T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 - 7.01T + 41T^{2} \) |
| 43 | \( 1 - 7.86T + 43T^{2} \) |
| 47 | \( 1 - 2.96T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 9.92T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 0.683T + 67T^{2} \) |
| 71 | \( 1 + 3.53T + 71T^{2} \) |
| 73 | \( 1 - 0.407T + 73T^{2} \) |
| 79 | \( 1 - 7.28T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.80452579579060155060282644353, −7.45352653711558250578624010935, −6.52571432279369548904526881768, −5.77109712612254622740352961694, −5.36085089733899590334804972438, −4.11711298645048061117821976669, −3.07269041421224312029660906665, −2.42547993265815295911299553356, −2.06195749525812658896100631395, −0.955507814206067275387571053669,
0.955507814206067275387571053669, 2.06195749525812658896100631395, 2.42547993265815295911299553356, 3.07269041421224312029660906665, 4.11711298645048061117821976669, 5.36085089733899590334804972438, 5.77109712612254622740352961694, 6.52571432279369548904526881768, 7.45352653711558250578624010935, 7.80452579579060155060282644353
