| L(s) = 1 | + 2-s − 0.764·3-s + 4-s − 0.445·5-s − 0.764·6-s − 0.938·7-s + 8-s − 2.41·9-s − 0.445·10-s + 2.33·11-s − 0.764·12-s − 6.49·13-s − 0.938·14-s + 0.340·15-s + 16-s + 1.93·17-s − 2.41·18-s − 19-s − 0.445·20-s + 0.717·21-s + 2.33·22-s + 3.50·23-s − 0.764·24-s − 4.80·25-s − 6.49·26-s + 4.13·27-s − 0.938·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.441·3-s + 0.5·4-s − 0.199·5-s − 0.311·6-s − 0.354·7-s + 0.353·8-s − 0.805·9-s − 0.140·10-s + 0.705·11-s − 0.220·12-s − 1.80·13-s − 0.250·14-s + 0.0879·15-s + 0.250·16-s + 0.470·17-s − 0.569·18-s − 0.229·19-s − 0.0996·20-s + 0.156·21-s + 0.498·22-s + 0.730·23-s − 0.155·24-s − 0.960·25-s − 1.27·26-s + 0.796·27-s − 0.177·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.782217746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.782217746\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 0.764T + 3T^{2} \) |
| 5 | \( 1 + 0.445T + 5T^{2} \) |
| 7 | \( 1 + 0.938T + 7T^{2} \) |
| 11 | \( 1 - 2.33T + 11T^{2} \) |
| 13 | \( 1 + 6.49T + 13T^{2} \) |
| 17 | \( 1 - 1.93T + 17T^{2} \) |
| 23 | \( 1 - 3.50T + 23T^{2} \) |
| 29 | \( 1 + 1.31T + 29T^{2} \) |
| 31 | \( 1 - 4.83T + 31T^{2} \) |
| 37 | \( 1 - 2.27T + 37T^{2} \) |
| 41 | \( 1 + 8.03T + 41T^{2} \) |
| 43 | \( 1 + 2.86T + 43T^{2} \) |
| 47 | \( 1 - 2.15T + 47T^{2} \) |
| 53 | \( 1 - 3.08T + 53T^{2} \) |
| 59 | \( 1 - 1.71T + 59T^{2} \) |
| 61 | \( 1 - 2.46T + 61T^{2} \) |
| 67 | \( 1 - 4.61T + 67T^{2} \) |
| 71 | \( 1 - 8.01T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 0.375T + 79T^{2} \) |
| 83 | \( 1 - 6.21T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−7.76163885780367253700751630940, −6.80606493365228872997493935137, −6.56220572553182516583737499023, −5.56466952073145280160419670315, −5.13089758714433051990516891196, −4.39279828599621922641242124761, −3.51190347696440592495138546847, −2.81588366492008312100704428287, −1.98347729837650387447684636666, −0.58003162821411874244693176842,
0.58003162821411874244693176842, 1.98347729837650387447684636666, 2.81588366492008312100704428287, 3.51190347696440592495138546847, 4.39279828599621922641242124761, 5.13089758714433051990516891196, 5.56466952073145280160419670315, 6.56220572553182516583737499023, 6.80606493365228872997493935137, 7.76163885780367253700751630940
