| L(s) = 1 | + 0.0557·2-s − 1.99·4-s + 3.85·5-s − 0.441·7-s − 0.222·8-s + 0.214·10-s + 11-s + 0.753·13-s − 0.0246·14-s + 3.98·16-s − 4.62·17-s + 1.94·19-s − 7.69·20-s + 0.0557·22-s − 4.57·23-s + 9.84·25-s + 0.0420·26-s + 0.881·28-s − 6.43·29-s − 8.50·31-s + 0.668·32-s − 0.258·34-s − 1.70·35-s − 2.46·37-s + 0.108·38-s − 0.859·40-s − 0.878·41-s + ⋯ |
| L(s) = 1 | + 0.0394·2-s − 0.998·4-s + 1.72·5-s − 0.166·7-s − 0.0788·8-s + 0.0679·10-s + 0.301·11-s + 0.208·13-s − 0.00658·14-s + 0.995·16-s − 1.12·17-s + 0.446·19-s − 1.72·20-s + 0.0118·22-s − 0.954·23-s + 1.96·25-s + 0.00824·26-s + 0.166·28-s − 1.19·29-s − 1.52·31-s + 0.118·32-s − 0.0442·34-s − 0.287·35-s − 0.405·37-s + 0.0176·38-s − 0.135·40-s − 0.137·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 0.0557T + 2T^{2} \) |
| 5 | \( 1 - 3.85T + 5T^{2} \) |
| 7 | \( 1 + 0.441T + 7T^{2} \) |
| 13 | \( 1 - 0.753T + 13T^{2} \) |
| 17 | \( 1 + 4.62T + 17T^{2} \) |
| 19 | \( 1 - 1.94T + 19T^{2} \) |
| 23 | \( 1 + 4.57T + 23T^{2} \) |
| 29 | \( 1 + 6.43T + 29T^{2} \) |
| 31 | \( 1 + 8.50T + 31T^{2} \) |
| 37 | \( 1 + 2.46T + 37T^{2} \) |
| 41 | \( 1 + 0.878T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 - 6.11T + 47T^{2} \) |
| 53 | \( 1 + 8.63T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 67 | \( 1 - 5.71T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 2.39T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 8.84T + 83T^{2} \) |
| 89 | \( 1 + 4.49T + 89T^{2} \) |
| 97 | \( 1 - 4.95T + 97T^{2} \) |
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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
−7.79281267560347645027381484148, −6.82112824116707637010179354888, −6.19990602669834824924508589855, −5.49459176663125262450622861308, −5.05065854527787190474070190025, −4.05788240972726889041851358830, −3.30245267100261820857522343464, −2.11699741862413277295312612041, −1.49448594013559927405454723350, 0,
1.49448594013559927405454723350, 2.11699741862413277295312612041, 3.30245267100261820857522343464, 4.05788240972726889041851358830, 5.05065854527787190474070190025, 5.49459176663125262450622861308, 6.19990602669834824924508589855, 6.82112824116707637010179354888, 7.79281267560347645027381484148
