| L(s) = 1 | − 2.33·2-s + 2.05·3-s + 3.46·4-s − 5-s − 4.79·6-s − 7-s − 3.43·8-s + 1.21·9-s + 2.33·10-s + 1.69·11-s + 7.11·12-s + 5.85·13-s + 2.33·14-s − 2.05·15-s + 1.08·16-s − 3.40·17-s − 2.83·18-s + 0.879·19-s − 3.46·20-s − 2.05·21-s − 3.95·22-s + 1.48·23-s − 7.04·24-s + 25-s − 13.6·26-s − 3.66·27-s − 3.46·28-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.18·3-s + 1.73·4-s − 0.447·5-s − 1.95·6-s − 0.377·7-s − 1.21·8-s + 0.404·9-s + 0.739·10-s + 0.510·11-s + 2.05·12-s + 1.62·13-s + 0.624·14-s − 0.529·15-s + 0.272·16-s − 0.825·17-s − 0.668·18-s + 0.201·19-s − 0.775·20-s − 0.447·21-s − 0.843·22-s + 0.310·23-s − 1.43·24-s + 0.200·25-s − 2.68·26-s − 0.705·27-s − 0.655·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 + 2.33T + 2T^{2} \) |
| 3 | \( 1 - 2.05T + 3T^{2} \) |
| 11 | \( 1 - 1.69T + 11T^{2} \) |
| 13 | \( 1 - 5.85T + 13T^{2} \) |
| 17 | \( 1 + 3.40T + 17T^{2} \) |
| 19 | \( 1 - 0.879T + 19T^{2} \) |
| 23 | \( 1 - 1.48T + 23T^{2} \) |
| 29 | \( 1 + 1.67T + 29T^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 + 3.02T + 37T^{2} \) |
| 41 | \( 1 - 2.80T + 41T^{2} \) |
| 43 | \( 1 + 4.67T + 43T^{2} \) |
| 47 | \( 1 + 9.01T + 47T^{2} \) |
| 53 | \( 1 - 9.83T + 53T^{2} \) |
| 59 | \( 1 - 2.90T + 59T^{2} \) |
| 61 | \( 1 + 8.22T + 61T^{2} \) |
| 67 | \( 1 + 7.60T + 67T^{2} \) |
| 71 | \( 1 - 7.95T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 + 6.90T + 79T^{2} \) |
| 83 | \( 1 - 3.98T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 5.01T + 97T^{2} \) |
| 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
−7.76212819121819648558413423461, −7.09874531293137683417401254632, −6.53116156629171304842349906704, −5.71055457014242014936031241665, −4.33675258732678768032605333771, −3.58533306887511823431797666434, −2.96545947326303982518906284176, −1.96605393929464332877679653705, −1.25921860031885685604057126377, 0,
1.25921860031885685604057126377, 1.96605393929464332877679653705, 2.96545947326303982518906284176, 3.58533306887511823431797666434, 4.33675258732678768032605333771, 5.71055457014242014936031241665, 6.53116156629171304842349906704, 7.09874531293137683417401254632, 7.76212819121819648558413423461
