| L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 10-s + 2·13-s + 4·14-s + 16-s + 6·17-s + 4·19-s − 20-s + 25-s + 2·26-s + 4·28-s + 6·29-s + 8·31-s + 32-s + 6·34-s − 4·35-s − 2·37-s + 4·38-s − 40-s + 6·41-s + 4·43-s + 9·49-s + 50-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 + 0.554·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.223·20-s + 1/5·25-s + 0.392·26-s + 0.755·28-s + 1.11·29-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 + 0.609·43-s + 9/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.456434411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.456434411\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.56562507638365, −14.11783443837333, −13.71264538609088, −13.13227769158495, −12.28370026801304, −12.02428439953040, −11.68699382876130, −10.97012637544470, −10.72453903162217, −9.968643852427164, −9.447020472108400, −8.483454844539392, −8.202287525112532, −7.701156031055550, −7.241268666852408, −6.444759953208820, −5.868186861375532, −5.175496384994778, −4.879872195131425, −4.193382393966589, −3.607924183484594, −2.936522099929823, −2.260409003977439, −1.199992779369645, −0.9938401776343899,
0.9938401776343899, 1.199992779369645, 2.260409003977439, 2.936522099929823, 3.607924183484594, 4.193382393966589, 4.879872195131425, 5.175496384994778, 5.868186861375532, 6.444759953208820, 7.241268666852408, 7.701156031055550, 8.202287525112532, 8.483454844539392, 9.447020472108400, 9.968643852427164, 10.72453903162217, 10.97012637544470, 11.68699382876130, 12.02428439953040, 12.28370026801304, 13.13227769158495, 13.71264538609088, 14.11783443837333, 14.56562507638365
