| L(s) = 1 | + 2-s − 4-s − 2·5-s + 7-s − 3·8-s − 2·10-s − 11-s + 14-s − 16-s + 4·17-s + 2·20-s − 22-s + 8·23-s − 25-s − 28-s + 5·32-s + 4·34-s − 2·35-s − 8·37-s + 6·40-s + 6·41-s − 4·43-s + 44-s + 8·46-s + 12·47-s + 49-s − 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s + 0.377·7-s − 1.06·8-s − 0.632·10-s − 0.301·11-s + 0.267·14-s − 1/4·16-s + 0.970·17-s + 0.447·20-s − 0.213·22-s + 1.66·23-s − 1/5·25-s − 0.188·28-s + 0.883·32-s + 0.685·34-s − 0.338·35-s − 1.31·37-s + 0.948·40-s + 0.937·41-s − 0.609·43-s + 0.150·44-s + 1.17·46-s + 1.75·47-s + 1/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.85895927642938, −13.41079122926931, −12.73986072243136, −12.45591078711311, −12.07777504306480, −11.49054363737658, −11.04714175325633, −10.57242838255226, −9.880879953348585, −9.444965330466949, −8.697060277876570, −8.583111114168062, −7.809762785629139, −7.433443955505820, −6.924365770208864, −6.165216269870928, −5.542082014599719, −5.205298480872580, −4.652694504176999, −4.100191029897804, −3.629270211611865, −3.060689209626370, −2.577204587144161, −1.496326265848473, −0.7981198822609304, 0,
0.7981198822609304, 1.496326265848473, 2.577204587144161, 3.060689209626370, 3.629270211611865, 4.100191029897804, 4.652694504176999, 5.205298480872580, 5.542082014599719, 6.165216269870928, 6.924365770208864, 7.433443955505820, 7.809762785629139, 8.583111114168062, 8.697060277876570, 9.444965330466949, 9.880879953348585, 10.57242838255226, 11.04714175325633, 11.49054363737658, 12.07777504306480, 12.45591078711311, 12.73986072243136, 13.41079122926931, 13.85895927642938
