| L(s) = 1 | + 2-s + 4-s + 5-s + 4·7-s + 8-s + 10-s + 11-s + 13-s + 4·14-s + 16-s − 6·17-s + 4·19-s + 20-s + 22-s + 4·23-s + 25-s + 26-s + 4·28-s + 2·29-s + 8·31-s + 32-s − 6·34-s + 4·35-s + 2·37-s + 4·38-s + 40-s − 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.51·7-s + 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.223·20-s + 0.213·22-s + 0.834·23-s + 1/5·25-s + 0.196·26-s + 0.755·28-s + 0.371·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s + 0.676·35-s + 0.328·37-s + 0.648·38-s + 0.158·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.324505622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.324505622\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−16.12606087190992, −15.53979036855571, −15.06881018056526, −14.55488726749392, −13.95452606425366, −13.53395053295058, −13.10784217981684, −12.23419987479200, −11.56747707901815, −11.37975241750107, −10.70377505723958, −10.06918636060317, −9.286168730984716, −8.556210168299383, −8.157100860208228, −7.321966032090689, −6.667933413774227, −6.156588214989422, −5.147600471611970, −4.903030781514029, −4.263978548357008, −3.333403712966153, −2.487542565160079, −1.750626106016043, −1.002165663369578,
1.002165663369578, 1.750626106016043, 2.487542565160079, 3.333403712966153, 4.263978548357008, 4.903030781514029, 5.147600471611970, 6.156588214989422, 6.667933413774227, 7.321966032090689, 8.157100860208228, 8.556210168299383, 9.286168730984716, 10.06918636060317, 10.70377505723958, 11.37975241750107, 11.56747707901815, 12.23419987479200, 13.10784217981684, 13.53395053295058, 13.95452606425366, 14.55488726749392, 15.06881018056526, 15.53979036855571, 16.12606087190992
