| L(s) = 1 | + 2-s + 4-s + 3·5-s − 7-s + 8-s + 3·10-s − 2·13-s − 14-s + 16-s − 17-s − 7·19-s + 3·20-s + 23-s + 4·25-s − 2·26-s − 28-s + 8·29-s − 9·31-s + 32-s − 34-s − 3·35-s − 5·37-s − 7·38-s + 3·40-s + 2·41-s + 8·43-s + 46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.377·7-s + 0.353·8-s + 0.948·10-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 1.60·19-s + 0.670·20-s + 0.208·23-s + 4/5·25-s − 0.392·26-s − 0.188·28-s + 1.48·29-s − 1.61·31-s + 0.176·32-s − 0.171·34-s − 0.507·35-s − 0.821·37-s − 1.13·38-s + 0.474·40-s + 0.312·41-s + 1.21·43-s + 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−13.06235775600582, −12.73154333049758, −12.22600754542598, −11.90586841345815, −11.08335005258875, −10.69420310486741, −10.35789260962363, −9.938451622067060, −9.352803323718859, −8.885039665497764, −8.568409248210643, −7.784874459357046, −7.216562282706908, −6.683539285020075, −6.464735611529590, −5.771893139165071, −5.520141766151819, −4.981635042661472, −4.272731362459338, −4.018549855942115, −3.154876060432785, −2.619332434666400, −2.181728790476819, −1.752235395789525, −0.9219801392654196, 0,
0.9219801392654196, 1.752235395789525, 2.181728790476819, 2.619332434666400, 3.154876060432785, 4.018549855942115, 4.272731362459338, 4.981635042661472, 5.520141766151819, 5.771893139165071, 6.464735611529590, 6.683539285020075, 7.216562282706908, 7.784874459357046, 8.568409248210643, 8.885039665497764, 9.352803323718859, 9.938451622067060, 10.35789260962363, 10.69420310486741, 11.08335005258875, 11.90586841345815, 12.22600754542598, 12.73154333049758, 13.06235775600582
