| L(s) = 1 | + 7.15·3-s + 8.30·5-s − 35.1·7-s + 24.1·9-s + 18.3·11-s + 40.0·13-s + 59.3·15-s − 125.·17-s − 19·19-s − 251.·21-s − 8.97·23-s − 56.0·25-s − 20.3·27-s − 153.·29-s + 114.·31-s + 130.·33-s − 291.·35-s − 83.5·37-s + 286.·39-s − 355.·41-s + 467.·43-s + 200.·45-s − 166.·47-s + 892.·49-s − 896.·51-s − 258.·53-s + 152·55-s + ⋯ |
| L(s) = 1 | + 1.37·3-s + 0.742·5-s − 1.89·7-s + 0.894·9-s + 0.501·11-s + 0.854·13-s + 1.02·15-s − 1.78·17-s − 0.229·19-s − 2.61·21-s − 0.0813·23-s − 0.448·25-s − 0.145·27-s − 0.982·29-s + 0.662·31-s + 0.690·33-s − 1.40·35-s − 0.371·37-s + 1.17·39-s − 1.35·41-s + 1.65·43-s + 0.664·45-s − 0.515·47-s + 2.60·49-s − 2.46·51-s − 0.669·53-s + 0.372·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 - 7.15T + 27T^{2} \) |
| 5 | \( 1 - 8.30T + 125T^{2} \) |
| 7 | \( 1 + 35.1T + 343T^{2} \) |
| 11 | \( 1 - 18.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 125.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 8.97T + 1.21e4T^{2} \) |
| 29 | \( 1 + 153.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 114.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 83.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 355.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 467.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 166.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 258.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 371.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 47.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 755.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 349.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 54.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + 438.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.07e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 501.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.43e3T + 9.12e5T^{2} \) |
| 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
−9.115591743301416199302857930347, −8.449458603357767068196691145450, −7.23296308135166431020012858388, −6.43675559353954060121423570727, −5.91827891666826328459040700990, −4.23292679634972068933911494695, −3.48102484450201908960288781950, −2.67356849369048313064078196400, −1.76574158876416607840065488590, 0,
1.76574158876416607840065488590, 2.67356849369048313064078196400, 3.48102484450201908960288781950, 4.23292679634972068933911494695, 5.91827891666826328459040700990, 6.43675559353954060121423570727, 7.23296308135166431020012858388, 8.449458603357767068196691145450, 9.115591743301416199302857930347
