| L(s) = 1 | + 3·3-s − 10.6·5-s + 9·9-s − 6.65·11-s + 75.9·13-s − 31.9·15-s − 104.·17-s + 85.4·19-s − 68.6·23-s − 11.4·25-s + 27·27-s + 87.7·29-s + 62.7·31-s − 19.9·33-s + 42.2·37-s + 227.·39-s + 313.·41-s + 306.·43-s − 95.8·45-s + 215.·47-s − 312.·51-s + 525.·53-s + 70.8·55-s + 256.·57-s + 360.·59-s + 800.·61-s − 809.·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.953·5-s + 0.333·9-s − 0.182·11-s + 1.62·13-s − 0.550·15-s − 1.48·17-s + 1.03·19-s − 0.622·23-s − 0.0917·25-s + 0.192·27-s + 0.562·29-s + 0.363·31-s − 0.105·33-s + 0.187·37-s + 0.935·39-s + 1.19·41-s + 1.08·43-s − 0.317·45-s + 0.667·47-s − 0.859·51-s + 1.36·53-s + 0.173·55-s + 0.595·57-s + 0.795·59-s + 1.68·61-s − 1.54·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.076349201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.076349201\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 10.6T + 125T^{2} \) |
| 11 | \( 1 + 6.65T + 1.33e3T^{2} \) |
| 13 | \( 1 - 75.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 85.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 68.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 87.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 62.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 42.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 313.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 306.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 215.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 525.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 360.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 800.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 40.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 298.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 517.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.22e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 639.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.42e3T + 9.12e5T^{2} \) |
| 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
−10.35833389526304273309073055564, −9.191305494877063274411994678401, −8.496038420823421550683751684965, −7.78131009041950979937325196402, −6.83956464623397615038187838264, −5.76659530442545319619885413370, −4.30848289549530939696449049009, −3.70771820213003901402818259259, −2.44211069902644343089517227232, −0.855674291971716201627547145745,
0.855674291971716201627547145745, 2.44211069902644343089517227232, 3.70771820213003901402818259259, 4.30848289549530939696449049009, 5.76659530442545319619885413370, 6.83956464623397615038187838264, 7.78131009041950979937325196402, 8.496038420823421550683751684965, 9.191305494877063274411994678401, 10.35833389526304273309073055564
