| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s + 25·5-s + 36·6-s − 83.0·7-s − 64·8-s + 81·9-s − 100·10-s + 449.·11-s − 144·12-s + 830.·13-s + 332.·14-s − 225·15-s + 256·16-s + 301.·17-s − 324·18-s − 361·19-s + 400·20-s + 747.·21-s − 1.79e3·22-s + 1.12e3·23-s + 576·24-s + 625·25-s − 3.32e3·26-s − 729·27-s − 1.32e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.640·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.11·11-s − 0.288·12-s + 1.36·13-s + 0.453·14-s − 0.258·15-s + 0.250·16-s + 0.253·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.369·21-s − 0.791·22-s + 0.444·23-s + 0.204·24-s + 0.200·25-s − 0.963·26-s − 0.192·27-s − 0.320·28-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
| \(L(3)\) |
\(\approx\) |
\(1.497120284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.497120284\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 83.0T + 1.68e4T^{2} \) |
| 11 | \( 1 - 449.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 830.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 301.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.12e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.98e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.61e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.41e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.19e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.21e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 498.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.06e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.90e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.49e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.17e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.55e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.33e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.34e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.20e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.31e5T + 8.58e9T^{2} \) |
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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
−9.907475329747769539156795890760, −9.165569761735303068199883398624, −8.400469791769204109133171735824, −7.10331738168637576288710296804, −6.34320954165209881953724772223, −5.76860894341084710567639413895, −4.24695575886012614631928121458, −3.12506916872403253395533888980, −1.60288645069809449760570261555, −0.73672383452891211905332475737,
0.73672383452891211905332475737, 1.60288645069809449760570261555, 3.12506916872403253395533888980, 4.24695575886012614631928121458, 5.76860894341084710567639413895, 6.34320954165209881953724772223, 7.10331738168637576288710296804, 8.400469791769204109133171735824, 9.165569761735303068199883398624, 9.907475329747769539156795890760
