| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s + 25·5-s + 36·6-s + 226.·7-s + 64·8-s + 81·9-s + 100·10-s + 619.·11-s + 144·12-s + 602.·13-s + 905.·14-s + 225·15-s + 256·16-s + 1.56e3·17-s + 324·18-s − 361·19-s + 400·20-s + 2.03e3·21-s + 2.47e3·22-s − 3.48e3·23-s + 576·24-s + 625·25-s + 2.41e3·26-s + 729·27-s + 3.62e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 1.74·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.54·11-s + 0.288·12-s + 0.989·13-s + 1.23·14-s + 0.258·15-s + 0.250·16-s + 1.31·17-s + 0.235·18-s − 0.229·19-s + 0.223·20-s + 1.00·21-s + 1.09·22-s − 1.37·23-s + 0.204·24-s + 0.200·25-s + 0.699·26-s + 0.192·27-s + 0.873·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.117598731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.117598731\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 - 9T \) |
| 5 | \( 1 - 25T \) |
| 19 | \( 1 + 361T \) |
| good | 7 | \( 1 - 226.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 619.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 602.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.56e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.48e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.88e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.08e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.36e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.90e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.70e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.72e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.05e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.41e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.28e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.21e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.50e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.03e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.74e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.31e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.05e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.01435906245983813359004337994, −8.915036593328788107214950454509, −8.191699238192545586858727725629, −7.30418305391254197366027007603, −6.15052126146782923184967430653, −5.29785949580221864184400327512, −4.18090291457127874610453021495, −3.46554055068234560609562293893, −1.73863262682445513638141680558, −1.48807313577180536354337616967,
1.48807313577180536354337616967, 1.73863262682445513638141680558, 3.46554055068234560609562293893, 4.18090291457127874610453021495, 5.29785949580221864184400327512, 6.15052126146782923184967430653, 7.30418305391254197366027007603, 8.191699238192545586858727725629, 8.915036593328788107214950454509, 10.01435906245983813359004337994
