| L(s) = 1 | + 0.656i·3-s + 0.656i·7-s + 2.56·9-s − 0.343·11-s + 1.91i·13-s + 4.48i·17-s + 19-s − 0.431·21-s + 3.56i·23-s + 3.65i·27-s − 7.99·29-s + 5.73·31-s − 0.225i·33-s − 4.16i·37-s − 1.25·39-s + ⋯ |
| L(s) = 1 | + 0.379i·3-s + 0.248i·7-s + 0.856·9-s − 0.103·11-s + 0.530i·13-s + 1.08i·17-s + 0.229·19-s − 0.0940·21-s + 0.744i·23-s + 0.703i·27-s − 1.48·29-s + 1.03·31-s − 0.0392i·33-s − 0.685i·37-s − 0.201·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.587021148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.587021148\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.656iT - 3T^{2} \) |
| 7 | \( 1 - 0.656iT - 7T^{2} \) |
| 11 | \( 1 + 0.343T + 11T^{2} \) |
| 13 | \( 1 - 1.91iT - 13T^{2} \) |
| 17 | \( 1 - 4.48iT - 17T^{2} \) |
| 23 | \( 1 - 3.56iT - 23T^{2} \) |
| 29 | \( 1 + 7.99T + 29T^{2} \) |
| 31 | \( 1 - 5.73T + 31T^{2} \) |
| 37 | \( 1 + 4.16iT - 37T^{2} \) |
| 41 | \( 1 + 9.08T + 41T^{2} \) |
| 43 | \( 1 + 3.51iT - 43T^{2} \) |
| 47 | \( 1 - 3.40iT - 47T^{2} \) |
| 53 | \( 1 - 1.68iT - 53T^{2} \) |
| 59 | \( 1 - 7.82T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 9.48iT - 67T^{2} \) |
| 71 | \( 1 + 9.96T + 71T^{2} \) |
| 73 | \( 1 + 7.53iT - 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 10.9iT - 83T^{2} \) |
| 89 | \( 1 - 4.19T + 89T^{2} \) |
| 97 | \( 1 - 1.73iT - 97T^{2} \) |
| 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
−8.922112117753794363410118227625, −7.935185370146756678814597324560, −7.32592192071415454664632087420, −6.52144299507814607279830446923, −5.71464057947804517782289088387, −4.95004410902713635860150955305, −4.05181969328075046486303025632, −3.52026849561204917171305787522, −2.21941951284794359741144319074, −1.34930806915059113116715711415,
0.47451669956612729669134553388, 1.57440518974004092979092703762, 2.63909753318428756730844710575, 3.55297171408828834017597842906, 4.53496222725466273279326686745, 5.16466178950363103819083318308, 6.15930253904647669945888421527, 6.90396263227852816107272314130, 7.47634757340038166537035919684, 8.112964885168464244104191690574
