| L(s) = 1 | − 4·2-s − 3·3-s + 16·4-s − 25·5-s + 12·6-s + 49·7-s − 64·8-s − 234·9-s + 100·10-s + 405·11-s − 48·12-s − 391·13-s − 196·14-s + 75·15-s + 256·16-s + 999·17-s + 936·18-s + 2.34e3·19-s − 400·20-s − 147·21-s − 1.62e3·22-s + 2.43e3·23-s + 192·24-s + 625·25-s + 1.56e3·26-s + 1.43e3·27-s + 784·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.192·3-s + 1/2·4-s − 0.447·5-s + 0.136·6-s + 0.377·7-s − 0.353·8-s − 0.962·9-s + 0.316·10-s + 1.00·11-s − 0.0962·12-s − 0.641·13-s − 0.267·14-s + 0.0860·15-s + 1/4·16-s + 0.838·17-s + 0.680·18-s + 1.48·19-s − 0.223·20-s − 0.0727·21-s − 0.713·22-s + 0.957·23-s + 0.0680·24-s + 1/5·25-s + 0.453·26-s + 0.377·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.096048321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.096048321\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + p T + p^{5} T^{2} \) |
| 11 | \( 1 - 405 T + p^{5} T^{2} \) |
| 13 | \( 1 + 391 T + p^{5} T^{2} \) |
| 17 | \( 1 - 999 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2342 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2430 T + p^{5} T^{2} \) |
| 29 | \( 1 - 8259 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4016 T + p^{5} T^{2} \) |
| 37 | \( 1 + 7042 T + p^{5} T^{2} \) |
| 41 | \( 1 - 3336 T + p^{5} T^{2} \) |
| 43 | \( 1 + 23518 T + p^{5} T^{2} \) |
| 47 | \( 1 - 10317 T + p^{5} T^{2} \) |
| 53 | \( 1 - 3084 T + p^{5} T^{2} \) |
| 59 | \( 1 + 18816 T + p^{5} T^{2} \) |
| 61 | \( 1 - 21668 T + p^{5} T^{2} \) |
| 67 | \( 1 - 52124 T + p^{5} T^{2} \) |
| 71 | \( 1 + 28560 T + p^{5} T^{2} \) |
| 73 | \( 1 + 70342 T + p^{5} T^{2} \) |
| 79 | \( 1 - 58823 T + p^{5} T^{2} \) |
| 83 | \( 1 - 756 T + p^{5} T^{2} \) |
| 89 | \( 1 - 135384 T + p^{5} T^{2} \) |
| 97 | \( 1 - 110435 T + p^{5} T^{2} \) |
| 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
−13.97933213951106159561516746547, −11.99914720514346147990752974246, −11.68138886583285412599376801300, −10.28329822660175891822872951645, −9.046943501853935919699138770096, −7.978038871649115483458270256611, −6.72786336549849558267719552520, −5.13838557355691162669440161510, −3.10672120100807710749122678147, −0.954505039202664782651900352722,
0.954505039202664782651900352722, 3.10672120100807710749122678147, 5.13838557355691162669440161510, 6.72786336549849558267719552520, 7.978038871649115483458270256611, 9.046943501853935919699138770096, 10.28329822660175891822872951645, 11.68138886583285412599376801300, 11.99914720514346147990752974246, 13.97933213951106159561516746547
