| L(s) = 1 | − 2-s − 0.381·3-s + 4-s − 0.381·5-s + 0.381·6-s − 1.85·7-s − 8-s − 2.85·9-s + 0.381·10-s − 11-s − 0.381·12-s − 5.85·13-s + 1.85·14-s + 0.145·15-s + 16-s + 6.47·17-s + 2.85·18-s − 0.381·20-s + 0.708·21-s + 22-s + 4.76·23-s + 0.381·24-s − 4.85·25-s + 5.85·26-s + 2.23·27-s − 1.85·28-s + 8.09·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.220·3-s + 0.5·4-s − 0.170·5-s + 0.155·6-s − 0.700·7-s − 0.353·8-s − 0.951·9-s + 0.120·10-s − 0.301·11-s − 0.110·12-s − 1.62·13-s + 0.495·14-s + 0.0376·15-s + 0.250·16-s + 1.56·17-s + 0.672·18-s − 0.0854·20-s + 0.154·21-s + 0.213·22-s + 0.993·23-s + 0.0779·24-s − 0.970·25-s + 1.14·26-s + 0.430·27-s − 0.350·28-s + 1.50·29-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.381T + 3T^{2} \) |
| 5 | \( 1 + 0.381T + 5T^{2} \) |
| 7 | \( 1 + 1.85T + 7T^{2} \) |
| 13 | \( 1 + 5.85T + 13T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 23 | \( 1 - 4.76T + 23T^{2} \) |
| 29 | \( 1 - 8.09T + 29T^{2} \) |
| 31 | \( 1 + 0.145T + 31T^{2} \) |
| 37 | \( 1 + 1.23T + 37T^{2} \) |
| 41 | \( 1 + 1.09T + 41T^{2} \) |
| 43 | \( 1 + 2.38T + 43T^{2} \) |
| 47 | \( 1 - 9.23T + 47T^{2} \) |
| 53 | \( 1 + 6.76T + 53T^{2} \) |
| 59 | \( 1 - 4.94T + 59T^{2} \) |
| 61 | \( 1 - 7.70T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 7.14T + 71T^{2} \) |
| 73 | \( 1 - 9.23T + 73T^{2} \) |
| 79 | \( 1 + 3.23T + 79T^{2} \) |
| 83 | \( 1 - 0.0901T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
| 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
−7.63323363181706286977588634429, −6.88006258807569534217758765166, −6.24770479091098462793458189517, −5.37691403989580382759315546853, −4.96889482335108073732989104262, −3.66182980426072345471026057119, −2.92158841943222128827299743767, −2.36000701515430917735766223140, −0.937974993016543774786215951119, 0,
0.937974993016543774786215951119, 2.36000701515430917735766223140, 2.92158841943222128827299743767, 3.66182980426072345471026057119, 4.96889482335108073732989104262, 5.37691403989580382759315546853, 6.24770479091098462793458189517, 6.88006258807569534217758765166, 7.63323363181706286977588634429
