L(s) = 1 | − 2-s − 3-s + 4-s + 0.962·5-s + 6-s + 0.389·7-s − 8-s + 9-s − 0.962·10-s + 3.00·11-s − 12-s + 5.33·13-s − 0.389·14-s − 0.962·15-s + 16-s − 17-s − 18-s − 5.20·19-s + 0.962·20-s − 0.389·21-s − 3.00·22-s − 4.24·23-s + 24-s − 4.07·25-s − 5.33·26-s − 27-s + 0.389·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.430·5-s + 0.408·6-s + 0.147·7-s − 0.353·8-s + 0.333·9-s − 0.304·10-s + 0.905·11-s − 0.288·12-s + 1.48·13-s − 0.104·14-s − 0.248·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s − 1.19·19-s + 0.215·20-s − 0.0849·21-s − 0.640·22-s − 0.884·23-s + 0.204·24-s − 0.814·25-s − 1.04·26-s − 0.192·27-s + 0.0735·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 - 0.962T + 5T^{2} \) |
| 7 | \( 1 - 0.389T + 7T^{2} \) |
| 11 | \( 1 - 3.00T + 11T^{2} \) |
| 13 | \( 1 - 5.33T + 13T^{2} \) |
| 19 | \( 1 + 5.20T + 19T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 - 7.55T + 29T^{2} \) |
| 31 | \( 1 - 0.492T + 31T^{2} \) |
| 37 | \( 1 + 2.63T + 37T^{2} \) |
| 41 | \( 1 + 2.24T + 41T^{2} \) |
| 43 | \( 1 + 6.00T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 61 | \( 1 - 5.36T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 5.11T + 71T^{2} \) |
| 73 | \( 1 + 4.42T + 73T^{2} \) |
| 79 | \( 1 + 7.70T + 79T^{2} \) |
| 83 | \( 1 + 4.47T + 83T^{2} \) |
| 89 | \( 1 + 8.41T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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
−7.948801154428505919622414064596, −6.75197969229751728560615103219, −6.35975415821973704717677154106, −5.96806789606175323538547727146, −4.84582003024591210298751216133, −4.06426533590560609443516714963, −3.20803595390808615238773362952, −1.86609125410200127584443712360, −1.38261318764869142319225984165, 0,
1.38261318764869142319225984165, 1.86609125410200127584443712360, 3.20803595390808615238773362952, 4.06426533590560609443516714963, 4.84582003024591210298751216133, 5.96806789606175323538547727146, 6.35975415821973704717677154106, 6.75197969229751728560615103219, 7.948801154428505919622414064596