| L(s) = 1 | − 6.50·3-s + 1.74·5-s + 26.0·7-s + 15.3·9-s − 34.5·11-s − 43.9·13-s − 11.3·15-s + 134.·17-s + 39.5·19-s − 169.·21-s − 22.5·23-s − 121.·25-s + 75.9·27-s + 29·29-s − 90.1·31-s + 224.·33-s + 45.4·35-s − 213.·37-s + 286.·39-s − 328.·41-s − 2.01·43-s + 26.7·45-s + 16.3·47-s + 336.·49-s − 873.·51-s − 149.·53-s − 60.2·55-s + ⋯ |
| L(s) = 1 | − 1.25·3-s + 0.156·5-s + 1.40·7-s + 0.567·9-s − 0.946·11-s − 0.938·13-s − 0.195·15-s + 1.91·17-s + 0.477·19-s − 1.76·21-s − 0.204·23-s − 0.975·25-s + 0.541·27-s + 0.185·29-s − 0.522·31-s + 1.18·33-s + 0.219·35-s − 0.947·37-s + 1.17·39-s − 1.25·41-s − 0.00715·43-s + 0.0885·45-s + 0.0507·47-s + 0.979·49-s − 2.39·51-s − 0.387·53-s − 0.147·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 6.50T + 27T^{2} \) |
| 5 | \( 1 - 1.74T + 125T^{2} \) |
| 7 | \( 1 - 26.0T + 343T^{2} \) |
| 11 | \( 1 + 34.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 43.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 134.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 22.5T + 1.21e4T^{2} \) |
| 31 | \( 1 + 90.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 213.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 328.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 2.01T + 7.95e4T^{2} \) |
| 47 | \( 1 - 16.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 149.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 292.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 111.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 159.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 129.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 742.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.06e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 496.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 676.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 103.T + 9.12e5T^{2} \) |
| 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
−8.141310670372908046590305917642, −7.80064201382713373753152481955, −6.91809386300579201820333311746, −5.73698528808703531217440596993, −5.23820002608457101374180643747, −4.86913546618819418628114940928, −3.49713246530950918263764098921, −2.18114626002615823487314471843, −1.14417621345180984456502250694, 0,
1.14417621345180984456502250694, 2.18114626002615823487314471843, 3.49713246530950918263764098921, 4.86913546618819418628114940928, 5.23820002608457101374180643747, 5.73698528808703531217440596993, 6.91809386300579201820333311746, 7.80064201382713373753152481955, 8.141310670372908046590305917642
