| L(s) = 1 | + 2-s − 4-s − 5-s − 7-s − 3·8-s − 3·9-s − 10-s − 2·11-s − 2·13-s − 14-s − 16-s − 2·17-s − 3·18-s − 2·19-s + 20-s − 2·22-s + 4·23-s + 25-s − 2·26-s + 28-s − 8·29-s + 5·32-s − 2·34-s + 35-s + 3·36-s + 2·37-s − 2·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.377·7-s − 1.06·8-s − 9-s − 0.316·10-s − 0.603·11-s − 0.554·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s − 0.707·18-s − 0.458·19-s + 0.223·20-s − 0.426·22-s + 0.834·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s − 1.48·29-s + 0.883·32-s − 0.342·34-s + 0.169·35-s + 1/2·36-s + 0.328·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58835 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58835 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.55845799994486, −14.15059487252242, −13.48901697009273, −12.95129902291237, −12.83516395967149, −12.18147704197733, −11.49956589750063, −11.26997589971109, −10.58895622509766, −9.964583888626469, −9.323014913786199, −8.913060605354546, −8.487308531550421, −7.794520721636328, −7.290916996932379, −6.606978450648725, −5.888955319116836, −5.583482121302314, −4.922854378612476, −4.404173471363618, −3.829751565611626, −3.066195899020672, −2.790854985418813, −1.947028589645532, −0.6137128915292472, 0,
0.6137128915292472, 1.947028589645532, 2.790854985418813, 3.066195899020672, 3.829751565611626, 4.404173471363618, 4.922854378612476, 5.583482121302314, 5.888955319116836, 6.606978450648725, 7.290916996932379, 7.794520721636328, 8.487308531550421, 8.913060605354546, 9.323014913786199, 9.964583888626469, 10.58895622509766, 11.26997589971109, 11.49956589750063, 12.18147704197733, 12.83516395967149, 12.95129902291237, 13.48901697009273, 14.15059487252242, 14.55845799994486
