| L(s) = 1 | + 2·2-s − 6.41·3-s + 4·4-s + 6.01·5-s − 12.8·6-s + 26.2·7-s + 8·8-s + 14.1·9-s + 12.0·10-s + 69.8·11-s − 25.6·12-s − 28.9·13-s + 52.5·14-s − 38.6·15-s + 16·16-s − 15.4·17-s + 28.3·18-s − 84.7·19-s + 24.0·20-s − 168.·21-s + 139.·22-s + 35.7·23-s − 51.3·24-s − 88.7·25-s − 57.9·26-s + 82.2·27-s + 105.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.23·3-s + 0.5·4-s + 0.538·5-s − 0.873·6-s + 1.41·7-s + 0.353·8-s + 0.525·9-s + 0.380·10-s + 1.91·11-s − 0.617·12-s − 0.618·13-s + 1.00·14-s − 0.664·15-s + 0.250·16-s − 0.221·17-s + 0.371·18-s − 1.02·19-s + 0.269·20-s − 1.75·21-s + 1.35·22-s + 0.324·23-s − 0.436·24-s − 0.710·25-s − 0.437·26-s + 0.585·27-s + 0.709·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.831574126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.831574126\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 269 | \( 1 - 269T \) |
| good | 3 | \( 1 + 6.41T + 27T^{2} \) |
| 5 | \( 1 - 6.01T + 125T^{2} \) |
| 7 | \( 1 - 26.2T + 343T^{2} \) |
| 11 | \( 1 - 69.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 28.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 15.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 84.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 35.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 95.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 189.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 76.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 387.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 562.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 469.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 424.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 490.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 564.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 111.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 717.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 328.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 557.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 873.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.05e3T + 7.04e5T^{2} \) |
| 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
−10.88144685885327027391919688022, −9.713784300066406088737294012044, −8.651883210101215581855232787263, −7.40139580584963469607197693867, −6.37343046911279419029154373859, −5.82332603079908902875583430299, −4.74989835915046747677367145062, −4.15335386323101858838104761138, −2.17992625302447889317828734785, −1.06770949401608027107714181934,
1.06770949401608027107714181934, 2.17992625302447889317828734785, 4.15335386323101858838104761138, 4.74989835915046747677367145062, 5.82332603079908902875583430299, 6.37343046911279419029154373859, 7.40139580584963469607197693867, 8.651883210101215581855232787263, 9.713784300066406088737294012044, 10.88144685885327027391919688022
