| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 7.31·5-s − 6·6-s + 17.3·7-s + 8·8-s + 9·9-s − 14.6·10-s + 69.2·11-s − 12·12-s − 3.25·13-s + 34.6·14-s + 21.9·15-s + 16·16-s + 85.9·17-s + 18·18-s − 19.9·19-s − 29.2·20-s − 51.9·21-s + 138.·22-s + 23·23-s − 24·24-s − 71.5·25-s − 6.50·26-s − 27·27-s + 69.2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.654·5-s − 0.408·6-s + 0.934·7-s + 0.353·8-s + 0.333·9-s − 0.462·10-s + 1.89·11-s − 0.288·12-s − 0.0694·13-s + 0.661·14-s + 0.377·15-s + 0.250·16-s + 1.22·17-s + 0.235·18-s − 0.240·19-s − 0.327·20-s − 0.539·21-s + 1.34·22-s + 0.208·23-s − 0.204·24-s − 0.572·25-s − 0.0491·26-s − 0.192·27-s + 0.467·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.261196599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.261196599\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 + 7.31T + 125T^{2} \) |
| 7 | \( 1 - 17.3T + 343T^{2} \) |
| 11 | \( 1 - 69.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 3.25T + 2.19e3T^{2} \) |
| 17 | \( 1 - 85.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 19.9T + 6.85e3T^{2} \) |
| 29 | \( 1 + 47.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 109.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 86.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 65.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 398.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 164.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 631.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 665.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 490.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 83.4T + 3.00e5T^{2} \) |
| 71 | \( 1 - 969.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 462.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 857.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.19e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 382.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 817.T + 9.12e5T^{2} \) |
| 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
−12.39478350542303761457797894326, −11.74913162801284593794002580579, −11.18440993382823306071536531247, −9.776743138095994712330981548737, −8.281076446293099021037767332728, −7.14585402216190036455212162180, −5.99467261876733706251465533603, −4.68803834745659696567350859198, −3.68932868994594777497651311234, −1.37973425137806912957340102785,
1.37973425137806912957340102785, 3.68932868994594777497651311234, 4.68803834745659696567350859198, 5.99467261876733706251465533603, 7.14585402216190036455212162180, 8.281076446293099021037767332728, 9.776743138095994712330981548737, 11.18440993382823306071536531247, 11.74913162801284593794002580579, 12.39478350542303761457797894326
