| L(s) = 1 | + 4.75·2-s + 3·3-s + 14.5·4-s + 14.2·6-s + 7·7-s + 31.3·8-s + 9·9-s + 7.31·11-s + 43.7·12-s − 4.15·13-s + 33.2·14-s + 32.2·16-s + 53.5·17-s + 42.7·18-s + 88.9·19-s + 21·21-s + 34.7·22-s + 156.·23-s + 94.0·24-s − 19.7·26-s + 27·27-s + 102.·28-s + 42.2·29-s − 14.0·31-s − 97.4·32-s + 21.9·33-s + 254.·34-s + ⋯ |
| L(s) = 1 | + 1.68·2-s + 0.577·3-s + 1.82·4-s + 0.970·6-s + 0.377·7-s + 1.38·8-s + 0.333·9-s + 0.200·11-s + 1.05·12-s − 0.0886·13-s + 0.635·14-s + 0.504·16-s + 0.763·17-s + 0.560·18-s + 1.07·19-s + 0.218·21-s + 0.337·22-s + 1.42·23-s + 0.799·24-s − 0.148·26-s + 0.192·27-s + 0.689·28-s + 0.270·29-s − 0.0812·31-s − 0.538·32-s + 0.115·33-s + 1.28·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.047871015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.047871015\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 4.75T + 8T^{2} \) |
| 11 | \( 1 - 7.31T + 1.33e3T^{2} \) |
| 13 | \( 1 + 4.15T + 2.19e3T^{2} \) |
| 17 | \( 1 - 53.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 88.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 156.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 42.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 14.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 293.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 127.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 210.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 468.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 115.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 314.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 768.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 717.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 737.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 477.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 279.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 776.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 29.7T + 7.04e5T^{2} \) |
| 97 | \( 1 + 231.T + 9.12e5T^{2} \) |
| 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
−10.73440536010869524680207651153, −9.644217557287299265420960903616, −8.552200895143457979645452056390, −7.42435159833599397914670702253, −6.68335271674406926938209006426, −5.42816531248477488642199228366, −4.81339307417873964966671028383, −3.59606247675145698222919911121, −2.91798430453661947416398156307, −1.51595956588549286391492374356,
1.51595956588549286391492374356, 2.91798430453661947416398156307, 3.59606247675145698222919911121, 4.81339307417873964966671028383, 5.42816531248477488642199228366, 6.68335271674406926938209006426, 7.42435159833599397914670702253, 8.552200895143457979645452056390, 9.644217557287299265420960903616, 10.73440536010869524680207651153
