| L(s) = 1 | + 2·2-s + 2·4-s + 7-s − 3·9-s − 6·11-s + 13-s + 2·14-s − 4·16-s − 4·17-s − 6·18-s + 5·19-s − 12·22-s − 3·23-s + 2·26-s + 2·28-s − 5·29-s − 3·31-s − 8·32-s − 8·34-s − 6·36-s + 4·37-s + 10·38-s − 6·41-s + 43-s − 12·44-s − 6·46-s − 7·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.377·7-s − 9-s − 1.80·11-s + 0.277·13-s + 0.534·14-s − 16-s − 0.970·17-s − 1.41·18-s + 1.14·19-s − 2.55·22-s − 0.625·23-s + 0.392·26-s + 0.377·28-s − 0.928·29-s − 0.538·31-s − 1.41·32-s − 1.37·34-s − 36-s + 0.657·37-s + 1.62·38-s − 0.937·41-s + 0.152·43-s − 1.80·44-s − 0.884·46-s − 1.02·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.481758772923047090659254291008, −7.80435801553106223064595186582, −6.90087579772522452258283983139, −5.83644097818684889229236969937, −5.41656791261413652587110709470, −4.77994485245253875660545017279, −3.73601074161005478747917475875, −2.89116428638083391976130354982, −2.15831794725409005407632083440, 0,
2.15831794725409005407632083440, 2.89116428638083391976130354982, 3.73601074161005478747917475875, 4.77994485245253875660545017279, 5.41656791261413652587110709470, 5.83644097818684889229236969937, 6.90087579772522452258283983139, 7.80435801553106223064595186582, 8.481758772923047090659254291008
