| L(s) = 1 | + 2-s + 3·3-s − 7·4-s + 5·5-s + 3·6-s − 7·7-s − 15·8-s + 9·9-s + 5·10-s + 11·11-s − 21·12-s + 54·13-s − 7·14-s + 15·15-s + 41·16-s − 78·17-s + 9·18-s − 148·19-s − 35·20-s − 21·21-s + 11·22-s + 200·23-s − 45·24-s + 25·25-s + 54·26-s + 27·27-s + 49·28-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 0.577·3-s − 7/8·4-s + 0.447·5-s + 0.204·6-s − 0.377·7-s − 0.662·8-s + 1/3·9-s + 0.158·10-s + 0.301·11-s − 0.505·12-s + 1.15·13-s − 0.133·14-s + 0.258·15-s + 0.640·16-s − 1.11·17-s + 0.117·18-s − 1.78·19-s − 0.391·20-s − 0.218·21-s + 0.106·22-s + 1.81·23-s − 0.382·24-s + 1/5·25-s + 0.407·26-s + 0.192·27-s + 0.330·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| 11 | \( 1 - p T \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 13 | \( 1 - 54 T + p^{3} T^{2} \) |
| 17 | \( 1 + 78 T + p^{3} T^{2} \) |
| 19 | \( 1 + 148 T + p^{3} T^{2} \) |
| 23 | \( 1 - 200 T + p^{3} T^{2} \) |
| 29 | \( 1 + 218 T + p^{3} T^{2} \) |
| 31 | \( 1 + 304 T + p^{3} T^{2} \) |
| 37 | \( 1 - 126 T + p^{3} T^{2} \) |
| 41 | \( 1 - 58 T + p^{3} T^{2} \) |
| 43 | \( 1 - 532 T + p^{3} T^{2} \) |
| 47 | \( 1 + 368 T + p^{3} T^{2} \) |
| 53 | \( 1 - 222 T + p^{3} T^{2} \) |
| 59 | \( 1 + 204 T + p^{3} T^{2} \) |
| 61 | \( 1 + 666 T + p^{3} T^{2} \) |
| 67 | \( 1 + 356 T + p^{3} T^{2} \) |
| 71 | \( 1 - 312 T + p^{3} T^{2} \) |
| 73 | \( 1 + 726 T + p^{3} T^{2} \) |
| 79 | \( 1 + 512 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1332 T + p^{3} T^{2} \) |
| 89 | \( 1 + 742 T + p^{3} T^{2} \) |
| 97 | \( 1 - 514 T + p^{3} T^{2} \) |
| 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
−9.000866879031951303160567551467, −8.579468400791720246868259787322, −7.31435690271869458741265315422, −6.35397284822721297530665512238, −5.62900817617123114296556038512, −4.42662521137849270210139395564, −3.83638976562542243480633613717, −2.77515608032814179534855275286, −1.50627436658102450005220850978, 0,
1.50627436658102450005220850978, 2.77515608032814179534855275286, 3.83638976562542243480633613717, 4.42662521137849270210139395564, 5.62900817617123114296556038512, 6.35397284822721297530665512238, 7.31435690271869458741265315422, 8.579468400791720246868259787322, 9.000866879031951303160567551467
