| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s − 10.0·5-s + 216·6-s + 971.·7-s + 512·8-s + 729·9-s − 80.5·10-s − 2.06e3·11-s + 1.72e3·12-s + 9.53e3·13-s + 7.76e3·14-s − 271.·15-s + 4.09e3·16-s − 2.08e4·17-s + 5.83e3·18-s + 4.10e4·19-s − 644.·20-s + 2.62e4·21-s − 1.65e4·22-s − 1.21e4·23-s + 1.38e4·24-s − 7.80e4·25-s + 7.62e4·26-s + 1.96e4·27-s + 6.21e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.0360·5-s + 0.408·6-s + 1.07·7-s + 0.353·8-s + 0.333·9-s − 0.0254·10-s − 0.468·11-s + 0.288·12-s + 1.20·13-s + 0.756·14-s − 0.0207·15-s + 0.250·16-s − 1.02·17-s + 0.235·18-s + 1.37·19-s − 0.0180·20-s + 0.617·21-s − 0.331·22-s − 0.208·23-s + 0.204·24-s − 0.998·25-s + 0.851·26-s + 0.192·27-s + 0.535·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.782560852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.782560852\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 3 | \( 1 - 27T \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 5 | \( 1 + 10.0T + 7.81e4T^{2} \) |
| 7 | \( 1 - 971.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.06e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 9.53e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.08e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.10e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 2.46e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.44e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.20e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.23e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.42e3T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.38e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.01e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.93e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.36e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.19e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.49e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.48e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.80e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.77e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.39e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.04e7T + 8.07e13T^{2} \) |
| 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
−11.73807394350758679866697110661, −11.13496336057619076193984703502, −9.839549731751206488044019576977, −8.433469927212164061211520789675, −7.70633699156495431572487492622, −6.29134115546031098007449671901, −4.99547553315877963488143416291, −3.92244545264270325534484283473, −2.55535905929611009953205279110, −1.25749473667986984974774172860,
1.25749473667986984974774172860, 2.55535905929611009953205279110, 3.92244545264270325534484283473, 4.99547553315877963488143416291, 6.29134115546031098007449671901, 7.70633699156495431572487492622, 8.433469927212164061211520789675, 9.839549731751206488044019576977, 11.13496336057619076193984703502, 11.73807394350758679866697110661
