| L(s) = 1 | + 2-s − 4-s − 2·5-s − 3·8-s − 2·10-s − 4·11-s − 13-s − 16-s + 17-s + 4·19-s + 2·20-s − 4·22-s − 25-s − 26-s + 2·29-s + 8·31-s + 5·32-s + 34-s − 2·37-s + 4·38-s + 6·40-s + 2·41-s − 4·43-s + 4·44-s + 8·47-s − 50-s + 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s − 0.632·10-s − 1.20·11-s − 0.277·13-s − 1/4·16-s + 0.242·17-s + 0.917·19-s + 0.447·20-s − 0.852·22-s − 1/5·25-s − 0.196·26-s + 0.371·29-s + 1.43·31-s + 0.883·32-s + 0.171·34-s − 0.328·37-s + 0.648·38-s + 0.948·40-s + 0.312·41-s − 0.609·43-s + 0.603·44-s + 1.16·47-s − 0.141·50-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.398673721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.398673721\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.69160784998694, −13.44424007641807, −12.77841721777142, −12.34586448287869, −11.87486987451510, −11.64003537186510, −10.88959609590332, −10.24185195915717, −9.974950204151524, −9.301210642182273, −8.703226144957044, −8.239963231001737, −7.741231319492237, −7.353916158708218, −6.658694576974186, −5.907498267470411, −5.514117526269426, −4.920811555357698, −4.514205223085029, −3.945505665013343, −3.310997302640097, −2.871052326682088, −2.257190680806315, −1.055091077182486, −0.3863036133917956,
0.3863036133917956, 1.055091077182486, 2.257190680806315, 2.871052326682088, 3.310997302640097, 3.945505665013343, 4.514205223085029, 4.920811555357698, 5.514117526269426, 5.907498267470411, 6.658694576974186, 7.353916158708218, 7.741231319492237, 8.239963231001737, 8.703226144957044, 9.301210642182273, 9.974950204151524, 10.24185195915717, 10.88959609590332, 11.64003537186510, 11.87486987451510, 12.34586448287869, 12.77841721777142, 13.44424007641807, 13.69160784998694
