L(s) = 1 | − 2-s + 4-s − 5-s − 4·7-s − 8-s + 10-s + 11-s + 13-s + 4·14-s + 16-s + 6·17-s − 4·19-s − 20-s − 22-s + 25-s − 26-s − 4·28-s + 6·29-s − 4·31-s − 32-s − 6·34-s + 4·35-s + 2·37-s + 4·38-s + 40-s + 6·41-s − 4·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s − 0.223·20-s − 0.213·22-s + 1/5·25-s − 0.196·26-s − 0.755·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 1.02·34-s + 0.676·35-s + 0.328·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8413358796\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8413358796\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−16.48246828360342, −15.77628346138738, −15.38159726240380, −14.67170437231359, −14.12068329414112, −13.32208920759921, −12.75440349831566, −12.20269160060603, −11.87553742122930, −10.86633985539304, −10.53100449683640, −9.823972699527472, −9.391632071903364, −8.767304386589642, −8.119883517215831, −7.467742237246035, −6.868940278239574, −6.184477063052261, −5.825873086894794, −4.704008329366774, −3.819542477159719, −3.249799995592812, −2.610964452875049, −1.421285817659045, −0.4806467155302907,
0.4806467155302907, 1.421285817659045, 2.610964452875049, 3.249799995592812, 3.819542477159719, 4.704008329366774, 5.825873086894794, 6.184477063052261, 6.868940278239574, 7.467742237246035, 8.119883517215831, 8.767304386589642, 9.391632071903364, 9.823972699527472, 10.53100449683640, 10.86633985539304, 11.87553742122930, 12.20269160060603, 12.75440349831566, 13.32208920759921, 14.12068329414112, 14.67170437231359, 15.38159726240380, 15.77628346138738, 16.48246828360342