| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 5·5-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s − 10·10-s + 11·11-s − 12·12-s + 30·13-s + 14·14-s + 15·15-s + 16·16-s − 5·17-s + 18·18-s − 109·19-s − 20·20-s − 21·21-s + 22·22-s − 199·23-s − 24·24-s + 25·25-s + 60·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.640·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.0713·17-s + 0.235·18-s − 1.31·19-s − 0.223·20-s − 0.218·21-s + 0.213·22-s − 1.80·23-s − 0.204·24-s + 1/5·25-s + 0.452·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| 11 | \( 1 - p T \) |
| good | 13 | \( 1 - 30 T + p^{3} T^{2} \) |
| 17 | \( 1 + 5 T + p^{3} T^{2} \) |
| 19 | \( 1 + 109 T + p^{3} T^{2} \) |
| 23 | \( 1 + 199 T + p^{3} T^{2} \) |
| 29 | \( 1 - 243 T + p^{3} T^{2} \) |
| 31 | \( 1 - 24 T + p^{3} T^{2} \) |
| 37 | \( 1 + 134 T + p^{3} T^{2} \) |
| 41 | \( 1 - 84 T + p^{3} T^{2} \) |
| 43 | \( 1 + 13 T + p^{3} T^{2} \) |
| 47 | \( 1 - 4 T + p^{3} T^{2} \) |
| 53 | \( 1 + 491 T + p^{3} T^{2} \) |
| 59 | \( 1 + 201 T + p^{3} T^{2} \) |
| 61 | \( 1 - 757 T + p^{3} T^{2} \) |
| 67 | \( 1 + 554 T + p^{3} T^{2} \) |
| 71 | \( 1 + 44 T + p^{3} T^{2} \) |
| 73 | \( 1 + 436 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1298 T + p^{3} T^{2} \) |
| 83 | \( 1 - 3 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1573 T + p^{3} T^{2} \) |
| 97 | \( 1 - 837 T + p^{3} T^{2} \) |
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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
−8.206339741059895773786006508962, −7.39274589480488613089298656137, −6.34128140778303477470160315951, −6.11680378888835808803180196020, −4.92456047909558737009625138159, −4.30278783877377292134266954054, −3.60460851930277711813397121964, −2.33969240751031640676626620929, −1.31494977199024235493373833449, 0,
1.31494977199024235493373833449, 2.33969240751031640676626620929, 3.60460851930277711813397121964, 4.30278783877377292134266954054, 4.92456047909558737009625138159, 6.11680378888835808803180196020, 6.34128140778303477470160315951, 7.39274589480488613089298656137, 8.206339741059895773786006508962
