| L(s) = 1 | − 2.73·3-s − 5-s + 4.46·9-s − 2·11-s − 0.732·13-s + 2.73·15-s + 7.46·17-s − 19-s + 1.46·23-s + 25-s − 3.99·27-s − 3.46·29-s − 4·31-s + 5.46·33-s + 7.66·37-s + 2·39-s + 0.535·41-s + 2.92·43-s − 4.46·45-s + 2.92·47-s − 7·49-s − 20.3·51-s − 11.6·53-s + 2·55-s + 2.73·57-s − 1.46·59-s − 6.53·61-s + ⋯ |
| L(s) = 1 | − 1.57·3-s − 0.447·5-s + 1.48·9-s − 0.603·11-s − 0.203·13-s + 0.705·15-s + 1.81·17-s − 0.229·19-s + 0.305·23-s + 0.200·25-s − 0.769·27-s − 0.643·29-s − 0.718·31-s + 0.951·33-s + 1.25·37-s + 0.320·39-s + 0.0836·41-s + 0.446·43-s − 0.665·45-s + 0.427·47-s − 49-s − 2.85·51-s − 1.60·53-s + 0.269·55-s + 0.361·57-s − 0.190·59-s − 0.836·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 0.732T + 13T^{2} \) |
| 17 | \( 1 - 7.46T + 17T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 7.66T + 37T^{2} \) |
| 41 | \( 1 - 0.535T + 41T^{2} \) |
| 43 | \( 1 - 2.92T + 43T^{2} \) |
| 47 | \( 1 - 2.92T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 1.46T + 59T^{2} \) |
| 61 | \( 1 + 6.53T + 61T^{2} \) |
| 67 | \( 1 - 4.19T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 8.39T + 79T^{2} \) |
| 83 | \( 1 - 5.46T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
| 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
−9.285903543570249034794321054606, −7.933432632411593856442196436643, −7.49175366085395270835962915034, −6.46669004041538687425617206179, −5.67706180849518134015040380342, −5.11122108738810566184958474401, −4.17999410036667507571703733933, −3.00674315700933097112064998749, −1.29533520211090669339640356869, 0,
1.29533520211090669339640356869, 3.00674315700933097112064998749, 4.17999410036667507571703733933, 5.11122108738810566184958474401, 5.67706180849518134015040380342, 6.46669004041538687425617206179, 7.49175366085395270835962915034, 7.933432632411593856442196436643, 9.285903543570249034794321054606
