| L(s) = 1 | + 0.729·2-s − 0.737·3-s − 7.46·4-s + 5·5-s − 0.537·6-s − 19.3·7-s − 11.2·8-s − 26.4·9-s + 3.64·10-s + 5.50·12-s − 19.1·13-s − 14.0·14-s − 3.68·15-s + 51.5·16-s + 34.8·17-s − 19.2·18-s − 37.2·19-s − 37.3·20-s + 14.2·21-s − 114.·23-s + 8.31·24-s + 25·25-s − 13.9·26-s + 39.4·27-s + 144.·28-s − 26.9·29-s − 2.68·30-s + ⋯ |
| L(s) = 1 | + 0.257·2-s − 0.141·3-s − 0.933·4-s + 0.447·5-s − 0.0365·6-s − 1.04·7-s − 0.498·8-s − 0.979·9-s + 0.115·10-s + 0.132·12-s − 0.408·13-s − 0.269·14-s − 0.0634·15-s + 0.805·16-s + 0.496·17-s − 0.252·18-s − 0.449·19-s − 0.417·20-s + 0.148·21-s − 1.03·23-s + 0.0707·24-s + 0.200·25-s − 0.105·26-s + 0.281·27-s + 0.974·28-s − 0.172·29-s − 0.0163·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9803097205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9803097205\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.729T + 8T^{2} \) |
| 3 | \( 1 + 0.737T + 27T^{2} \) |
| 7 | \( 1 + 19.3T + 343T^{2} \) |
| 13 | \( 1 + 19.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 34.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 37.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 26.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 45.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 331.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 241.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 285.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 72.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 317.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 755.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 908.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 626.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 968.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 459.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 637.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 756.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 513.T + 9.12e5T^{2} \) |
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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
−9.964585743796890086819644104062, −9.518988908894267115805392350865, −8.637265136601379075430266097107, −7.69641029234519182974141479293, −6.22217705834303858157865877081, −5.82525836234253868763578602045, −4.68743252325151150921151787028, −3.57159616363893212170914201098, −2.54995377727790515653594079328, −0.55107762226854394798171265955,
0.55107762226854394798171265955, 2.54995377727790515653594079328, 3.57159616363893212170914201098, 4.68743252325151150921151787028, 5.82525836234253868763578602045, 6.22217705834303858157865877081, 7.69641029234519182974141479293, 8.637265136601379075430266097107, 9.518988908894267115805392350865, 9.964585743796890086819644104062
