| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 2·11-s − 6·13-s + 16-s + 2·17-s − 6·19-s + 20-s − 2·22-s + 4·23-s + 25-s + 6·26-s − 6·29-s − 32-s − 2·34-s + 6·37-s + 6·38-s − 40-s + 4·41-s + 2·44-s − 4·46-s − 4·47-s − 7·49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 0.603·11-s − 1.66·13-s + 1/4·16-s + 0.485·17-s − 1.37·19-s + 0.223·20-s − 0.426·22-s + 0.834·23-s + 1/5·25-s + 1.17·26-s − 1.11·29-s − 0.176·32-s − 0.342·34-s + 0.986·37-s + 0.973·38-s − 0.158·40-s + 0.624·41-s + 0.301·44-s − 0.589·46-s − 0.583·47-s − 49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.365779248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.365779248\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.06368194240325, −12.75753066340613, −12.29845494187593, −11.76258154388630, −11.23216358182561, −10.90111256314666, −10.19135505366779, −9.870335246633200, −9.477726954432568, −9.011026796872444, −8.489014164940465, −7.985701569044360, −7.352696717972051, −7.019641182852930, −6.559623658866286, −5.874209327526185, −5.493487569151490, −4.788394280179229, −4.311598136349365, −3.626999376384665, −2.895641838135202, −2.326559218246743, −1.935190821791968, −1.127767544466630, −0.4019396194164819,
0.4019396194164819, 1.127767544466630, 1.935190821791968, 2.326559218246743, 2.895641838135202, 3.626999376384665, 4.311598136349365, 4.788394280179229, 5.493487569151490, 5.874209327526185, 6.559623658866286, 7.019641182852930, 7.352696717972051, 7.985701569044360, 8.489014164940465, 9.011026796872444, 9.477726954432568, 9.870335246633200, 10.19135505366779, 10.90111256314666, 11.23216358182561, 11.76258154388630, 12.29845494187593, 12.75753066340613, 13.06368194240325
