| L(s) = 1 | + 2-s + 4-s − 3.46·5-s + 7-s + 8-s − 3.46·10-s − 11-s + 2·13-s + 14-s + 16-s + 3.46·17-s − 1.46·19-s − 3.46·20-s − 22-s + 6.92·23-s + 6.99·25-s + 2·26-s + 28-s + 6·29-s − 1.46·31-s + 32-s + 3.46·34-s − 3.46·35-s + 8.92·37-s − 1.46·38-s − 3.46·40-s + 3.46·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.54·5-s + 0.377·7-s + 0.353·8-s − 1.09·10-s − 0.301·11-s + 0.554·13-s + 0.267·14-s + 0.250·16-s + 0.840·17-s − 0.335·19-s − 0.774·20-s − 0.213·22-s + 1.44·23-s + 1.39·25-s + 0.392·26-s + 0.188·28-s + 1.11·29-s − 0.262·31-s + 0.176·32-s + 0.594·34-s − 0.585·35-s + 1.46·37-s − 0.237·38-s − 0.547·40-s + 0.541·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.143676183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.143676183\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 1.46T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 - 8.92T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 2.92T + 43T^{2} \) |
| 47 | \( 1 + 2.53T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 1.07T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.584280973472457077532098453311, −8.440896253551250356218315482221, −7.923337773819737726126142632621, −7.19714710749690088289298432029, −6.28669996459717867188376423926, −5.16136149441923229926405952281, −4.43692597497406014978953032748, −3.60788259867162602760732647188, −2.76072760305783002965781308600, −0.998111015528653354260886153503,
0.998111015528653354260886153503, 2.76072760305783002965781308600, 3.60788259867162602760732647188, 4.43692597497406014978953032748, 5.16136149441923229926405952281, 6.28669996459717867188376423926, 7.19714710749690088289298432029, 7.923337773819737726126142632621, 8.440896253551250356218315482221, 9.584280973472457077532098453311
