| L(s) = 1 | + 3·2-s + 3·3-s + 4-s + 9·6-s − 7·7-s − 21·8-s + 9·9-s − 36·11-s + 3·12-s + 34·13-s − 21·14-s − 71·16-s − 42·17-s + 27·18-s − 124·19-s − 21·21-s − 108·22-s − 63·24-s + 102·26-s + 27·27-s − 7·28-s + 102·29-s − 160·31-s − 45·32-s − 108·33-s − 126·34-s + 9·36-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.577·3-s + 1/8·4-s + 0.612·6-s − 0.377·7-s − 0.928·8-s + 1/3·9-s − 0.986·11-s + 0.0721·12-s + 0.725·13-s − 0.400·14-s − 1.10·16-s − 0.599·17-s + 0.353·18-s − 1.49·19-s − 0.218·21-s − 1.04·22-s − 0.535·24-s + 0.769·26-s + 0.192·27-s − 0.0472·28-s + 0.653·29-s − 0.926·31-s − 0.248·32-s − 0.569·33-s − 0.635·34-s + 1/24·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 + 42 T + p^{3} T^{2} \) |
| 19 | \( 1 + 124 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 - 102 T + p^{3} T^{2} \) |
| 31 | \( 1 + 160 T + p^{3} T^{2} \) |
| 37 | \( 1 + 398 T + p^{3} T^{2} \) |
| 41 | \( 1 + 318 T + p^{3} T^{2} \) |
| 43 | \( 1 - 268 T + p^{3} T^{2} \) |
| 47 | \( 1 + 240 T + p^{3} T^{2} \) |
| 53 | \( 1 - 498 T + p^{3} T^{2} \) |
| 59 | \( 1 + 132 T + p^{3} T^{2} \) |
| 61 | \( 1 - 398 T + p^{3} T^{2} \) |
| 67 | \( 1 + 92 T + p^{3} T^{2} \) |
| 71 | \( 1 + 720 T + p^{3} T^{2} \) |
| 73 | \( 1 - 502 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1024 T + p^{3} T^{2} \) |
| 83 | \( 1 - 204 T + p^{3} T^{2} \) |
| 89 | \( 1 - 354 T + p^{3} T^{2} \) |
| 97 | \( 1 - 286 T + p^{3} T^{2} \) |
| 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
−10.12075963038195753167479392371, −8.857664619071613451766903593028, −8.462808045494271645908698043295, −7.05934179263858124393768043409, −6.15591888066082578281245494934, −5.13942924283493950264659608217, −4.14241923520507025757724574463, −3.26655394703988871667371446388, −2.17378914361499586968822560904, 0,
2.17378914361499586968822560904, 3.26655394703988871667371446388, 4.14241923520507025757724574463, 5.13942924283493950264659608217, 6.15591888066082578281245494934, 7.05934179263858124393768043409, 8.462808045494271645908698043295, 8.857664619071613451766903593028, 10.12075963038195753167479392371
