| L(s) = 1 | − 2·2-s + 4·4-s + 14·5-s + 7·7-s − 8·8-s − 28·10-s + 11·11-s − 16·13-s − 14·14-s + 16·16-s − 108·17-s + 116·19-s + 56·20-s − 22·22-s − 68·23-s + 71·25-s + 32·26-s + 28·28-s − 122·29-s − 262·31-s − 32·32-s + 216·34-s + 98·35-s + 130·37-s − 232·38-s − 112·40-s − 204·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.25·5-s + 0.377·7-s − 0.353·8-s − 0.885·10-s + 0.301·11-s − 0.341·13-s − 0.267·14-s + 1/4·16-s − 1.54·17-s + 1.40·19-s + 0.626·20-s − 0.213·22-s − 0.616·23-s + 0.567·25-s + 0.241·26-s + 0.188·28-s − 0.781·29-s − 1.51·31-s − 0.176·32-s + 1.08·34-s + 0.473·35-s + 0.577·37-s − 0.990·38-s − 0.442·40-s − 0.777·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| 11 | \( 1 - p T \) |
| good | 5 | \( 1 - 14 T + p^{3} T^{2} \) |
| 13 | \( 1 + 16 T + p^{3} T^{2} \) |
| 17 | \( 1 + 108 T + p^{3} T^{2} \) |
| 19 | \( 1 - 116 T + p^{3} T^{2} \) |
| 23 | \( 1 + 68 T + p^{3} T^{2} \) |
| 29 | \( 1 + 122 T + p^{3} T^{2} \) |
| 31 | \( 1 + 262 T + p^{3} T^{2} \) |
| 37 | \( 1 - 130 T + p^{3} T^{2} \) |
| 41 | \( 1 + 204 T + p^{3} T^{2} \) |
| 43 | \( 1 + 396 T + p^{3} T^{2} \) |
| 47 | \( 1 + 166 T + p^{3} T^{2} \) |
| 53 | \( 1 + 442 T + p^{3} T^{2} \) |
| 59 | \( 1 + 702 T + p^{3} T^{2} \) |
| 61 | \( 1 - 196 T + p^{3} T^{2} \) |
| 67 | \( 1 + 416 T + p^{3} T^{2} \) |
| 71 | \( 1 + 492 T + p^{3} T^{2} \) |
| 73 | \( 1 - 408 T + p^{3} T^{2} \) |
| 79 | \( 1 - 600 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1212 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1146 T + p^{3} T^{2} \) |
| 97 | \( 1 + 482 T + p^{3} T^{2} \) |
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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
−9.102070693262266012995891336730, −8.062882904890862625651410012621, −7.20634538434498624830327614029, −6.40389452810723188627414594486, −5.60687993451256688531131076496, −4.73811548226100006340100167694, −3.34855889017840624999712561016, −2.10789422559820437720030141443, −1.55296442445375211958552898971, 0,
1.55296442445375211958552898971, 2.10789422559820437720030141443, 3.34855889017840624999712561016, 4.73811548226100006340100167694, 5.60687993451256688531131076496, 6.40389452810723188627414594486, 7.20634538434498624830327614029, 8.062882904890862625651410012621, 9.102070693262266012995891336730
