| L(s) = 1 | − 2·2-s + 7·3-s + 4·4-s + 5·5-s − 14·6-s + 20·7-s − 8·8-s + 22·9-s − 10·10-s + 6·11-s + 28·12-s + 47·13-s − 40·14-s + 35·15-s + 16·16-s − 132·17-s − 44·18-s + 146·19-s + 20·20-s + 140·21-s − 12·22-s + 23·23-s − 56·24-s + 25·25-s − 94·26-s − 35·27-s + 80·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.34·3-s + 1/2·4-s + 0.447·5-s − 0.952·6-s + 1.07·7-s − 0.353·8-s + 0.814·9-s − 0.316·10-s + 0.164·11-s + 0.673·12-s + 1.00·13-s − 0.763·14-s + 0.602·15-s + 1/4·16-s − 1.88·17-s − 0.576·18-s + 1.76·19-s + 0.223·20-s + 1.45·21-s − 0.116·22-s + 0.208·23-s − 0.476·24-s + 1/5·25-s − 0.709·26-s − 0.249·27-s + 0.539·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.472707715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.472707715\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 23 | \( 1 - p T \) |
| good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 7 | \( 1 - 20 T + p^{3} T^{2} \) |
| 11 | \( 1 - 6 T + p^{3} T^{2} \) |
| 13 | \( 1 - 47 T + p^{3} T^{2} \) |
| 17 | \( 1 + 132 T + p^{3} T^{2} \) |
| 19 | \( 1 - 146 T + p^{3} T^{2} \) |
| 29 | \( 1 + 99 T + p^{3} T^{2} \) |
| 31 | \( 1 + 253 T + p^{3} T^{2} \) |
| 37 | \( 1 + 118 T + p^{3} T^{2} \) |
| 41 | \( 1 - 495 T + p^{3} T^{2} \) |
| 43 | \( 1 - 272 T + p^{3} T^{2} \) |
| 47 | \( 1 - 639 T + p^{3} T^{2} \) |
| 53 | \( 1 + 342 T + p^{3} T^{2} \) |
| 59 | \( 1 - 240 T + p^{3} T^{2} \) |
| 61 | \( 1 + 370 T + p^{3} T^{2} \) |
| 67 | \( 1 - 698 T + p^{3} T^{2} \) |
| 71 | \( 1 + 357 T + p^{3} T^{2} \) |
| 73 | \( 1 + 259 T + p^{3} T^{2} \) |
| 79 | \( 1 - 542 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1248 T + p^{3} T^{2} \) |
| 89 | \( 1 + 828 T + p^{3} T^{2} \) |
| 97 | \( 1 - 992 T + p^{3} T^{2} \) |
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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
−11.37077038113681963182557592306, −10.82444281105823496989068898624, −9.212075945774321871342889026596, −9.071279867799872453345479510799, −7.976394719154291149562640142669, −7.16335027409095941770976244763, −5.63166526039217538278909604408, −3.98698912411610601015955250214, −2.53871522810913860014886883857, −1.45467661833420215073522048284,
1.45467661833420215073522048284, 2.53871522810913860014886883857, 3.98698912411610601015955250214, 5.63166526039217538278909604408, 7.16335027409095941770976244763, 7.976394719154291149562640142669, 9.071279867799872453345479510799, 9.212075945774321871342889026596, 10.82444281105823496989068898624, 11.37077038113681963182557592306
