| L(s) = 1 | + 2·5-s − 7-s − 3·9-s − 4·11-s − 2·13-s − 6·17-s − 25-s − 6·29-s − 8·31-s − 2·35-s + 2·37-s − 2·41-s − 4·43-s − 6·45-s − 8·47-s + 49-s − 6·53-s − 8·55-s − 6·61-s + 3·63-s − 4·65-s + 4·67-s + 8·71-s + 10·73-s + 4·77-s − 16·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s − 9-s − 1.20·11-s − 0.554·13-s − 1.45·17-s − 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.338·35-s + 0.328·37-s − 0.312·41-s − 0.609·43-s − 0.894·45-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 1.07·55-s − 0.768·61-s + 0.377·63-s − 0.496·65-s + 0.488·67-s + 0.949·71-s + 1.17·73-s + 0.455·77-s − 1.80·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5267870366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5267870366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 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
−15.57670645977811, −15.14266196060220, −14.48983239353678, −14.02003719719551, −13.31733251348600, −13.07052654004478, −12.59029095893272, −11.68721574754380, −11.13482804567237, −10.77122850550435, −9.994889096072198, −9.541842258964907, −9.002070238773437, −8.411605898863995, −7.706379152093886, −7.133057276547134, −6.290124298844007, −5.938508305791338, −5.159107279082855, −4.846457737199814, −3.704471537134639, −3.042875665701969, −2.216950524040245, −1.926045745824907, −0.2704923647878087,
0.2704923647878087, 1.926045745824907, 2.216950524040245, 3.042875665701969, 3.704471537134639, 4.846457737199814, 5.159107279082855, 5.938508305791338, 6.290124298844007, 7.133057276547134, 7.706379152093886, 8.411605898863995, 9.002070238773437, 9.541842258964907, 9.994889096072198, 10.77122850550435, 11.13482804567237, 11.68721574754380, 12.59029095893272, 13.07052654004478, 13.31733251348600, 14.02003719719551, 14.48983239353678, 15.14266196060220, 15.57670645977811
