| L(s) = 1 | + 3·2-s + 5·3-s + 4-s + 15·6-s + 7·7-s − 21·8-s − 2·9-s − 25·11-s + 5·12-s + 13·13-s + 21·14-s − 71·16-s − 46·17-s − 6·18-s + 74·19-s + 35·21-s − 75·22-s + 55·23-s − 105·24-s + 39·26-s − 145·27-s + 7·28-s + 82·29-s + 325·31-s − 45·32-s − 125·33-s − 138·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.962·3-s + 1/8·4-s + 1.02·6-s + 0.377·7-s − 0.928·8-s − 0.0740·9-s − 0.685·11-s + 0.120·12-s + 0.277·13-s + 0.400·14-s − 1.10·16-s − 0.656·17-s − 0.0785·18-s + 0.893·19-s + 0.363·21-s − 0.726·22-s + 0.498·23-s − 0.893·24-s + 0.294·26-s − 1.03·27-s + 0.0472·28-s + 0.525·29-s + 1.88·31-s − 0.248·32-s − 0.659·33-s − 0.696·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.738504645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.738504645\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| 13 | \( 1 - p T \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 11 | \( 1 + 25 T + p^{3} T^{2} \) |
| 17 | \( 1 + 46 T + p^{3} T^{2} \) |
| 19 | \( 1 - 74 T + p^{3} T^{2} \) |
| 23 | \( 1 - 55 T + p^{3} T^{2} \) |
| 29 | \( 1 - 82 T + p^{3} T^{2} \) |
| 31 | \( 1 - 325 T + p^{3} T^{2} \) |
| 37 | \( 1 - 181 T + p^{3} T^{2} \) |
| 41 | \( 1 - 201 T + p^{3} T^{2} \) |
| 43 | \( 1 + 298 T + p^{3} T^{2} \) |
| 47 | \( 1 - 421 T + p^{3} T^{2} \) |
| 53 | \( 1 - 432 T + p^{3} T^{2} \) |
| 59 | \( 1 - 24 T + p^{3} T^{2} \) |
| 61 | \( 1 + 75 T + p^{3} T^{2} \) |
| 67 | \( 1 - 911 T + p^{3} T^{2} \) |
| 71 | \( 1 - 464 T + p^{3} T^{2} \) |
| 73 | \( 1 - 285 T + p^{3} T^{2} \) |
| 79 | \( 1 + 335 T + p^{3} T^{2} \) |
| 83 | \( 1 + 306 T + p^{3} T^{2} \) |
| 89 | \( 1 + 782 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1739 T + p^{3} T^{2} \) |
| 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
−8.530911377430246918987424660878, −8.088347796113695826640094221200, −7.08482137259614558346512528596, −6.12634509892147353209036406468, −5.33083412314536434081012421985, −4.60320791783086515794332600436, −3.78579523021781616949729237695, −2.87645346302510778456052268751, −2.38950587450273148191106841866, −0.78035668389662233637538272325,
0.78035668389662233637538272325, 2.38950587450273148191106841866, 2.87645346302510778456052268751, 3.78579523021781616949729237695, 4.60320791783086515794332600436, 5.33083412314536434081012421985, 6.12634509892147353209036406468, 7.08482137259614558346512528596, 8.088347796113695826640094221200, 8.530911377430246918987424660878
