| L(s) = 1 | + 1.93·2-s − 2.61·3-s + 1.74·4-s − 5-s − 5.05·6-s + 7-s − 0.494·8-s + 3.81·9-s − 1.93·10-s − 4.55·12-s + 2.11·13-s + 1.93·14-s + 2.61·15-s − 4.44·16-s − 5.18·17-s + 7.38·18-s + 5.34·19-s − 1.74·20-s − 2.61·21-s + 5.31·23-s + 1.28·24-s + 25-s + 4.09·26-s − 2.12·27-s + 1.74·28-s − 2.45·29-s + 5.05·30-s + ⋯ |
| L(s) = 1 | + 1.36·2-s − 1.50·3-s + 0.872·4-s − 0.447·5-s − 2.06·6-s + 0.377·7-s − 0.174·8-s + 1.27·9-s − 0.611·10-s − 1.31·12-s + 0.586·13-s + 0.517·14-s + 0.673·15-s − 1.11·16-s − 1.25·17-s + 1.73·18-s + 1.22·19-s − 0.390·20-s − 0.569·21-s + 1.10·23-s + 0.263·24-s + 0.200·25-s + 0.803·26-s − 0.408·27-s + 0.329·28-s − 0.455·29-s + 0.922·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.93T + 2T^{2} \) |
| 3 | \( 1 + 2.61T + 3T^{2} \) |
| 13 | \( 1 - 2.11T + 13T^{2} \) |
| 17 | \( 1 + 5.18T + 17T^{2} \) |
| 19 | \( 1 - 5.34T + 19T^{2} \) |
| 23 | \( 1 - 5.31T + 23T^{2} \) |
| 29 | \( 1 + 2.45T + 29T^{2} \) |
| 31 | \( 1 + 6.26T + 31T^{2} \) |
| 37 | \( 1 - 6.82T + 37T^{2} \) |
| 41 | \( 1 - 1.91T + 41T^{2} \) |
| 43 | \( 1 + 5.23T + 43T^{2} \) |
| 47 | \( 1 + 5.48T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 9.29T + 61T^{2} \) |
| 67 | \( 1 + 7.45T + 67T^{2} \) |
| 71 | \( 1 - 7.37T + 71T^{2} \) |
| 73 | \( 1 + 6.61T + 73T^{2} \) |
| 79 | \( 1 - 9.28T + 79T^{2} \) |
| 83 | \( 1 + 2.17T + 83T^{2} \) |
| 89 | \( 1 + 8.52T + 89T^{2} \) |
| 97 | \( 1 + 0.130T + 97T^{2} \) |
| 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
−7.67088796286120976802904732601, −6.91777081615080334982184512210, −6.33642542251758182280836347271, −5.59416671734659032931693776253, −5.07238780692023546950265618168, −4.47053574540703709676938498687, −3.73607862712437498669193997138, −2.76589504566288891801587903061, −1.33819516686197425488857813623, 0,
1.33819516686197425488857813623, 2.76589504566288891801587903061, 3.73607862712437498669193997138, 4.47053574540703709676938498687, 5.07238780692023546950265618168, 5.59416671734659032931693776253, 6.33642542251758182280836347271, 6.91777081615080334982184512210, 7.67088796286120976802904732601
