| L(s) = 1 | − 1.71·2-s + 3.18·3-s + 0.955·4-s − 4.00·5-s − 5.47·6-s − 0.822·7-s + 1.79·8-s + 7.14·9-s + 6.88·10-s − 11-s + 3.04·12-s − 6.89·13-s + 1.41·14-s − 12.7·15-s − 4.99·16-s + 17-s − 12.2·18-s + 4.72·19-s − 3.82·20-s − 2.61·21-s + 1.71·22-s − 3.39·23-s + 5.71·24-s + 11.0·25-s + 11.8·26-s + 13.1·27-s − 0.785·28-s + ⋯ |
| L(s) = 1 | − 1.21·2-s + 1.83·3-s + 0.477·4-s − 1.79·5-s − 2.23·6-s − 0.310·7-s + 0.634·8-s + 2.38·9-s + 2.17·10-s − 0.301·11-s + 0.878·12-s − 1.91·13-s + 0.377·14-s − 3.29·15-s − 1.24·16-s + 0.242·17-s − 2.89·18-s + 1.08·19-s − 0.856·20-s − 0.571·21-s + 0.366·22-s − 0.706·23-s + 1.16·24-s + 2.21·25-s + 2.32·26-s + 2.53·27-s − 0.148·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 1.71T + 2T^{2} \) |
| 3 | \( 1 - 3.18T + 3T^{2} \) |
| 5 | \( 1 + 4.00T + 5T^{2} \) |
| 7 | \( 1 + 0.822T + 7T^{2} \) |
| 13 | \( 1 + 6.89T + 13T^{2} \) |
| 19 | \( 1 - 4.72T + 19T^{2} \) |
| 23 | \( 1 + 3.39T + 23T^{2} \) |
| 29 | \( 1 - 8.69T + 29T^{2} \) |
| 31 | \( 1 + 8.91T + 31T^{2} \) |
| 37 | \( 1 - 5.10T + 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 47 | \( 1 - 8.72T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 3.72T + 61T^{2} \) |
| 67 | \( 1 - 7.77T + 67T^{2} \) |
| 71 | \( 1 + 3.49T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 7.74T + 79T^{2} \) |
| 83 | \( 1 + 5.25T + 83T^{2} \) |
| 89 | \( 1 + 4.17T + 89T^{2} \) |
| 97 | \( 1 - 8.41T + 97T^{2} \) |
| 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
−7.73446637136634829301109360048, −7.41813135992040472102781506565, −6.81381934695148425347650426574, −5.04744039392900166669038290645, −4.37251026885429412878363566595, −3.77352921986476222243651114792, −2.90570257199569801548526807794, −2.37838623194828261490744826025, −1.09678698217886216455819068335, 0,
1.09678698217886216455819068335, 2.37838623194828261490744826025, 2.90570257199569801548526807794, 3.77352921986476222243651114792, 4.37251026885429412878363566595, 5.04744039392900166669038290645, 6.81381934695148425347650426574, 7.41813135992040472102781506565, 7.73446637136634829301109360048
