| L(s) = 1 | + 5.38·2-s − 4.56·3-s + 20.9·4-s − 5·5-s − 24.5·6-s − 26.2·7-s + 69.7·8-s − 6.17·9-s − 26.9·10-s + 2.68·11-s − 95.6·12-s + 62.7·13-s − 141.·14-s + 22.8·15-s + 207.·16-s − 33.2·18-s − 119.·19-s − 104.·20-s + 119.·21-s + 14.4·22-s + 173.·23-s − 318.·24-s + 25·25-s + 337.·26-s + 151.·27-s − 550.·28-s − 117.·29-s + ⋯ |
| L(s) = 1 | + 1.90·2-s − 0.878·3-s + 2.61·4-s − 0.447·5-s − 1.67·6-s − 1.41·7-s + 3.08·8-s − 0.228·9-s − 0.850·10-s + 0.0735·11-s − 2.30·12-s + 1.33·13-s − 2.69·14-s + 0.392·15-s + 3.24·16-s − 0.435·18-s − 1.44·19-s − 1.17·20-s + 1.24·21-s + 0.139·22-s + 1.57·23-s − 2.70·24-s + 0.200·25-s + 2.54·26-s + 1.07·27-s − 3.71·28-s − 0.751·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.432396446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.432396446\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 5.38T + 8T^{2} \) |
| 3 | \( 1 + 4.56T + 27T^{2} \) |
| 7 | \( 1 + 26.2T + 343T^{2} \) |
| 11 | \( 1 - 2.68T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62.7T + 2.19e3T^{2} \) |
| 19 | \( 1 + 119.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 173.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 117.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 145.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 190.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 38.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 450.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 353.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 24.3T + 1.48e5T^{2} \) |
| 59 | \( 1 + 496.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 825.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 864.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 521.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 192.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 847.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 265.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 718.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 291.T + 9.12e5T^{2} \) |
| 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
−9.227887657325856007794671499925, −8.085676885297723195003283249420, −6.79094221669800660919747694668, −6.46250448833735111961061389277, −5.87955361345085514302755720993, −5.00181173938298349501035637660, −4.04424775103568744296316575262, −3.38719825522719364651300482243, −2.50514737139086232076545907483, −0.801025611064115405256060537452,
0.801025611064115405256060537452, 2.50514737139086232076545907483, 3.38719825522719364651300482243, 4.04424775103568744296316575262, 5.00181173938298349501035637660, 5.87955361345085514302755720993, 6.46250448833735111961061389277, 6.79094221669800660919747694668, 8.085676885297723195003283249420, 9.227887657325856007794671499925
