| L(s) = 1 | − 1.51·5-s + i·7-s − 0.935i·11-s + 2.14i·13-s − 2.62i·17-s + 3.46·19-s − 6.86·23-s − 2.70·25-s + 2.44·29-s − 2i·31-s − 1.51i·35-s − 2.14i·37-s − 2.62i·41-s + 7.74·43-s − 49-s + ⋯ |
| L(s) = 1 | − 0.677·5-s + 0.377i·7-s − 0.282i·11-s + 0.593i·13-s − 0.635i·17-s + 0.794·19-s − 1.43·23-s − 0.541·25-s + 0.454·29-s − 0.359i·31-s − 0.255i·35-s − 0.351i·37-s − 0.409i·41-s + 1.18·43-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 + 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.769 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.313461755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.313461755\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 + 1.51T + 5T^{2} \) |
| 11 | \( 1 + 0.935iT - 11T^{2} \) |
| 13 | \( 1 - 2.14iT - 13T^{2} \) |
| 17 | \( 1 + 2.62iT - 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + 6.86T + 23T^{2} \) |
| 29 | \( 1 - 2.44T + 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 + 2.14iT - 37T^{2} \) |
| 41 | \( 1 + 2.62iT - 41T^{2} \) |
| 43 | \( 1 - 7.74T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 9.79iT - 59T^{2} \) |
| 61 | \( 1 - 11.2iT - 61T^{2} \) |
| 67 | \( 1 + 2.14T + 67T^{2} \) |
| 71 | \( 1 + 1.62T + 71T^{2} \) |
| 73 | \( 1 + 5.70T + 73T^{2} \) |
| 79 | \( 1 - 1.70iT - 79T^{2} \) |
| 83 | \( 1 + 1.87iT - 83T^{2} \) |
| 89 | \( 1 - 2.62iT - 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−8.321649658579735526698362031587, −7.62261406455305750669619783829, −7.05612607191046131437865850661, −6.04823772723526625617356029264, −5.48940631044749585461164692241, −4.42867358418825092684369796245, −3.84854910362046742523400573919, −2.86761018616851718425397393790, −1.91756556501572744962387892512, −0.50476464160693743428662373347,
0.841276518142740889112856152390, 2.07229246632545079181621063676, 3.21103863959386440193113641840, 3.93752412080038698263470070993, 4.61797154307980365382762020242, 5.62275370171588646406478456288, 6.27236060122550872406781010111, 7.27908278036493515311108320067, 7.76315649440014730806193332557, 8.348075361817996552868813410657
