| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s − 25·5-s − 36·6-s + 154.·7-s − 64·8-s + 81·9-s + 100·10-s − 170.·11-s + 144·12-s + 428.·13-s − 616.·14-s − 225·15-s + 256·16-s + 643.·17-s − 324·18-s − 361·19-s − 400·20-s + 1.38e3·21-s + 683.·22-s + 603.·23-s − 576·24-s + 625·25-s − 1.71e3·26-s + 729·27-s + 2.46e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 1.18·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.425·11-s + 0.288·12-s + 0.703·13-s − 0.840·14-s − 0.258·15-s + 0.250·16-s + 0.540·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.686·21-s + 0.301·22-s + 0.237·23-s − 0.204·24-s + 0.200·25-s − 0.497·26-s + 0.192·27-s + 0.594·28-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
| \(L(3)\) |
\(\approx\) |
\(2.206365732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.206365732\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 154.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 170.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 428.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 643.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 603.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.79e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.51e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.65e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.00e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.82e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.23e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.87e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.05e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.70e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.38e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.43e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.01e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.55e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.33e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.83e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.83e5T + 8.58e9T^{2} \) |
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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
−9.934682163466102678144488832395, −8.848693179198303567888538339610, −8.163113146369949522307717537583, −7.70833859563672402158646913692, −6.60013416826559938788894970292, −5.31228658379674816771639260819, −4.21660383978034999830046830743, −3.02893916777086754668929426466, −1.84295369798531709677678349983, −0.820846785257774734501779757734,
0.820846785257774734501779757734, 1.84295369798531709677678349983, 3.02893916777086754668929426466, 4.21660383978034999830046830743, 5.31228658379674816771639260819, 6.60013416826559938788894970292, 7.70833859563672402158646913692, 8.163113146369949522307717537583, 8.848693179198303567888538339610, 9.934682163466102678144488832395
