| L(s) = 1 | + 2-s − 1.32·3-s + 4-s + 1.76·5-s − 1.32·6-s − 7-s + 8-s − 1.25·9-s + 1.76·10-s − 0.812·11-s − 1.32·12-s − 3.05·13-s − 14-s − 2.33·15-s + 16-s + 5.93·17-s − 1.25·18-s − 1.55·19-s + 1.76·20-s + 1.32·21-s − 0.812·22-s − 2.32·23-s − 1.32·24-s − 1.88·25-s − 3.05·26-s + 5.62·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.763·3-s + 0.5·4-s + 0.789·5-s − 0.539·6-s − 0.377·7-s + 0.353·8-s − 0.417·9-s + 0.558·10-s − 0.244·11-s − 0.381·12-s − 0.847·13-s − 0.267·14-s − 0.602·15-s + 0.250·16-s + 1.43·17-s − 0.295·18-s − 0.356·19-s + 0.394·20-s + 0.288·21-s − 0.173·22-s − 0.484·23-s − 0.269·24-s − 0.376·25-s − 0.599·26-s + 1.08·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + 1.32T + 3T^{2} \) |
| 5 | \( 1 - 1.76T + 5T^{2} \) |
| 11 | \( 1 + 0.812T + 11T^{2} \) |
| 13 | \( 1 + 3.05T + 13T^{2} \) |
| 17 | \( 1 - 5.93T + 17T^{2} \) |
| 19 | \( 1 + 1.55T + 19T^{2} \) |
| 23 | \( 1 + 2.32T + 23T^{2} \) |
| 29 | \( 1 + 3.27T + 29T^{2} \) |
| 31 | \( 1 - 6.76T + 31T^{2} \) |
| 37 | \( 1 + 4.26T + 37T^{2} \) |
| 41 | \( 1 - 5.53T + 41T^{2} \) |
| 43 | \( 1 + 9.60T + 43T^{2} \) |
| 47 | \( 1 - 6.16T + 47T^{2} \) |
| 53 | \( 1 + 2.51T + 53T^{2} \) |
| 59 | \( 1 + 3.99T + 59T^{2} \) |
| 61 | \( 1 + 9.38T + 61T^{2} \) |
| 67 | \( 1 - 8.20T + 67T^{2} \) |
| 71 | \( 1 + 3.77T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 - 5.26T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 4.13T + 89T^{2} \) |
| 97 | \( 1 + 1.97T + 97T^{2} \) |
| 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
−7.57583591959943946116266709314, −6.74190684618321713924960841215, −6.06184228010012818677318684361, −5.60475997586695600015087087230, −5.07510340616548020954001075718, −4.18996824027702453505311177364, −3.14669172914314223445839381576, −2.50275917618533844118930099021, −1.42114552542714497617010832004, 0,
1.42114552542714497617010832004, 2.50275917618533844118930099021, 3.14669172914314223445839381576, 4.18996824027702453505311177364, 5.07510340616548020954001075718, 5.60475997586695600015087087230, 6.06184228010012818677318684361, 6.74190684618321713924960841215, 7.57583591959943946116266709314
