| L(s) = 1 | + 2-s + 4-s − 2·5-s − 7-s + 8-s − 3·9-s − 2·10-s − 11-s − 14-s + 16-s + 2·17-s − 3·18-s + 4·19-s − 2·20-s − 22-s − 25-s − 28-s + 6·29-s − 8·31-s + 32-s + 2·34-s + 2·35-s − 3·36-s − 6·37-s + 4·38-s − 2·40-s − 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s + 0.353·8-s − 9-s − 0.632·10-s − 0.301·11-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.707·18-s + 0.917·19-s − 0.447·20-s − 0.213·22-s − 1/5·25-s − 0.188·28-s + 1.11·29-s − 1.43·31-s + 0.176·32-s + 0.342·34-s + 0.338·35-s − 1/2·36-s − 0.986·37-s + 0.648·38-s − 0.316·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.663023955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.663023955\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.36691984942216, −14.71732151713416, −14.14307135226010, −13.92036902612472, −13.14507231993867, −12.58246451700083, −12.03880742458014, −11.73236534185265, −11.11579255721435, −10.61012680205897, −9.987486230670161, −9.200613011674250, −8.702026447636685, −7.942270340818268, −7.533143980246213, −7.006168372421487, −6.103043366875105, −5.743629804018982, −5.034171890254418, −4.455203652483930, −3.535026896722163, −3.277419989083292, −2.594379643796868, −1.583514883840048, −0.4426707298346358,
0.4426707298346358, 1.583514883840048, 2.594379643796868, 3.277419989083292, 3.535026896722163, 4.455203652483930, 5.034171890254418, 5.743629804018982, 6.103043366875105, 7.006168372421487, 7.533143980246213, 7.942270340818268, 8.702026447636685, 9.200613011674250, 9.987486230670161, 10.61012680205897, 11.11579255721435, 11.73236534185265, 12.03880742458014, 12.58246451700083, 13.14507231993867, 13.92036902612472, 14.14307135226010, 14.71732151713416, 15.36691984942216
