| L(s) = 1 | + 2.64·2-s − 0.999·4-s − 10.5·5-s − 22·7-s − 23.8·8-s − 28.0·10-s − 5.29·11-s + 13·13-s − 58.2·14-s − 55.0·16-s + 116.·17-s − 126·19-s + 10.5·20-s − 14.0·22-s + 31.7·23-s − 12.9·25-s + 34.3·26-s + 21.9·28-s + 52.9·29-s − 182·31-s + 44.9·32-s + 308·34-s + 232.·35-s − 86·37-s − 333.·38-s + 252.·40-s − 444.·41-s + ⋯ |
| L(s) = 1 | + 0.935·2-s − 0.124·4-s − 0.946·5-s − 1.18·7-s − 1.05·8-s − 0.885·10-s − 0.145·11-s + 0.277·13-s − 1.11·14-s − 0.859·16-s + 1.66·17-s − 1.52·19-s + 0.118·20-s − 0.135·22-s + 0.287·23-s − 0.103·25-s + 0.259·26-s + 0.148·28-s + 0.338·29-s − 1.05·31-s + 0.248·32-s + 1.55·34-s + 1.12·35-s − 0.382·37-s − 1.42·38-s + 0.996·40-s − 1.69·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 - 2.64T + 8T^{2} \) |
| 5 | \( 1 + 10.5T + 125T^{2} \) |
| 7 | \( 1 + 22T + 343T^{2} \) |
| 11 | \( 1 + 5.29T + 1.33e3T^{2} \) |
| 17 | \( 1 - 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 126T + 6.85e3T^{2} \) |
| 23 | \( 1 - 31.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 52.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 182T + 2.97e4T^{2} \) |
| 37 | \( 1 + 86T + 5.06e4T^{2} \) |
| 41 | \( 1 + 444.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 96T + 7.95e4T^{2} \) |
| 47 | \( 1 - 365.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 190.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 587.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 574T + 2.26e5T^{2} \) |
| 67 | \( 1 + 530T + 3.00e5T^{2} \) |
| 71 | \( 1 - 809.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 154T + 3.89e5T^{2} \) |
| 79 | \( 1 + 460T + 4.93e5T^{2} \) |
| 83 | \( 1 + 322.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.43e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 70T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.57561237509761188203366248690, −11.98073174854902702505941289239, −10.54087326293509430015122423146, −9.319254954172655942930698156517, −8.134922373224238986393423304919, −6.69620929653497116695108899445, −5.52600227951355270372844370243, −4.04012168483176216528019367997, −3.20433354821540860461744264412, 0,
3.20433354821540860461744264412, 4.04012168483176216528019367997, 5.52600227951355270372844370243, 6.69620929653497116695108899445, 8.134922373224238986393423304919, 9.319254954172655942930698156517, 10.54087326293509430015122423146, 11.98073174854902702505941289239, 12.57561237509761188203366248690
