| L(s) = 1 | − 2·2-s + 4·4-s + 12·5-s + 7·7-s − 8·8-s − 24·10-s + 50·11-s − 13·13-s − 14·14-s + 16·16-s + 58·17-s − 40·19-s + 48·20-s − 100·22-s + 64·23-s + 19·25-s + 26·26-s + 28·28-s + 110·29-s + 124·31-s − 32·32-s − 116·34-s + 84·35-s − 50·37-s + 80·38-s − 96·40-s − 84·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.07·5-s + 0.377·7-s − 0.353·8-s − 0.758·10-s + 1.37·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.827·17-s − 0.482·19-s + 0.536·20-s − 0.969·22-s + 0.580·23-s + 0.151·25-s + 0.196·26-s + 0.188·28-s + 0.704·29-s + 0.718·31-s − 0.176·32-s − 0.585·34-s + 0.405·35-s − 0.222·37-s + 0.341·38-s − 0.379·40-s − 0.319·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.461188473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.461188473\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| 13 | \( 1 + p T \) |
| good | 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 50 T + p^{3} T^{2} \) |
| 17 | \( 1 - 58 T + p^{3} T^{2} \) |
| 19 | \( 1 + 40 T + p^{3} T^{2} \) |
| 23 | \( 1 - 64 T + p^{3} T^{2} \) |
| 29 | \( 1 - 110 T + p^{3} T^{2} \) |
| 31 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 37 | \( 1 + 50 T + p^{3} T^{2} \) |
| 41 | \( 1 + 84 T + p^{3} T^{2} \) |
| 43 | \( 1 + 12 T + p^{3} T^{2} \) |
| 47 | \( 1 - 82 T + p^{3} T^{2} \) |
| 53 | \( 1 - 442 T + p^{3} T^{2} \) |
| 59 | \( 1 - 618 T + p^{3} T^{2} \) |
| 61 | \( 1 + 278 T + p^{3} T^{2} \) |
| 67 | \( 1 - 20 T + p^{3} T^{2} \) |
| 71 | \( 1 - 390 T + p^{3} T^{2} \) |
| 73 | \( 1 + 2 T + p^{3} T^{2} \) |
| 79 | \( 1 + 680 T + p^{3} T^{2} \) |
| 83 | \( 1 + 322 T + p^{3} T^{2} \) |
| 89 | \( 1 + 968 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1022 T + p^{3} T^{2} \) |
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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
−8.979107139181802914279472595706, −8.491346640315099574551547128670, −7.41270392717720456475545357131, −6.62867995633400095719349312110, −5.96376053363551765129790756314, −5.04996051734227734403239141167, −3.90399395467728576403003254967, −2.68218841865384642607123064299, −1.69744418920316327205172004387, −0.901867459137294766704125628510,
0.901867459137294766704125628510, 1.69744418920316327205172004387, 2.68218841865384642607123064299, 3.90399395467728576403003254967, 5.04996051734227734403239141167, 5.96376053363551765129790756314, 6.62867995633400095719349312110, 7.41270392717720456475545357131, 8.491346640315099574551547128670, 8.979107139181802914279472595706
