| L(s) = 1 | − 8.11·3-s − 11.0·5-s − 5.74·7-s + 38.9·9-s − 34.5·11-s + 29.8·13-s + 89.5·15-s − 17·17-s − 115.·19-s + 46.6·21-s + 84.9·23-s − 3.34·25-s − 96.8·27-s + 144.·29-s + 24.1·31-s + 280.·33-s + 63.3·35-s + 221.·37-s − 242.·39-s + 345.·41-s + 540.·43-s − 429.·45-s + 354.·47-s − 309.·49-s + 138.·51-s − 66.6·53-s + 380.·55-s + ⋯ |
| L(s) = 1 | − 1.56·3-s − 0.986·5-s − 0.310·7-s + 1.44·9-s − 0.946·11-s + 0.637·13-s + 1.54·15-s − 0.242·17-s − 1.39·19-s + 0.484·21-s + 0.769·23-s − 0.0267·25-s − 0.690·27-s + 0.923·29-s + 0.139·31-s + 1.47·33-s + 0.306·35-s + 0.982·37-s − 0.995·39-s + 1.31·41-s + 1.91·43-s − 1.42·45-s + 1.10·47-s − 0.903·49-s + 0.378·51-s − 0.172·53-s + 0.933·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 3 | \( 1 + 8.11T + 27T^{2} \) |
| 5 | \( 1 + 11.0T + 125T^{2} \) |
| 7 | \( 1 + 5.74T + 343T^{2} \) |
| 11 | \( 1 + 34.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 29.8T + 2.19e3T^{2} \) |
| 19 | \( 1 + 115.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 84.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 144.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 24.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 221.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 345.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 540.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 354.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 66.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 611.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 623.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 730.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 566.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 937.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 707.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 306.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 707.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.30e3T + 9.12e5T^{2} \) |
| 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
−9.085425207961714796676964535856, −8.095717587789324293792816574740, −7.31428542822313756602529216515, −6.35661808595195352775281480099, −5.79576685253517940409433464682, −4.67746777634845719727334772852, −4.09594230353308603231494086220, −2.67012655883595908745782312872, −0.890746659622087687852580261028, 0,
0.890746659622087687852580261028, 2.67012655883595908745782312872, 4.09594230353308603231494086220, 4.67746777634845719727334772852, 5.79576685253517940409433464682, 6.35661808595195352775281480099, 7.31428542822313756602529216515, 8.095717587789324293792816574740, 9.085425207961714796676964535856
