L(s) = 1 | − 16·2-s + 76.8·3-s + 256·4-s + 1.09e3·5-s − 1.22e3·6-s − 4.09e3·8-s − 1.37e4·9-s − 1.74e4·10-s + 5.40e4·11-s + 1.96e4·12-s − 3.78e4·13-s + 8.39e4·15-s + 6.55e4·16-s + 3.53e5·17-s + 2.20e5·18-s − 5.02e4·19-s + 2.79e5·20-s − 8.64e5·22-s + 2.45e6·23-s − 3.14e5·24-s − 7.60e5·25-s + 6.04e5·26-s − 2.57e6·27-s + 1.40e6·29-s − 1.34e6·30-s − 5.78e6·31-s − 1.04e6·32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.547·3-s + 0.5·4-s + 0.781·5-s − 0.387·6-s − 0.353·8-s − 0.699·9-s − 0.552·10-s + 1.11·11-s + 0.273·12-s − 0.367·13-s + 0.428·15-s + 0.250·16-s + 1.02·17-s + 0.494·18-s − 0.0883·19-s + 0.390·20-s − 0.787·22-s + 1.83·23-s − 0.193·24-s − 0.389·25-s + 0.259·26-s − 0.931·27-s + 0.369·29-s − 0.302·30-s − 1.12·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(2.194859536\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.194859536\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 16T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 76.8T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.09e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 5.40e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.78e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.53e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.02e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.45e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.40e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.78e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 5.45e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.73e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.43e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.04e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.30e6T + 3.29e15T^{2} \) |
| 59 | \( 1 - 6.61e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.00e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.72e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.43e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.68e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.63e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 7.47e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.12e9T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.27e8T + 7.60e17T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.95137956321721932132607890787, −10.89938240932632903668083568738, −9.570910591085020315277297599625, −9.080455479241642329412342234498, −7.86069650027035754840828050147, −6.60324184833313543951395616652, −5.40667326701574640538640322595, −3.43354035348610912285936934605, −2.20207593793785912644808211339, −0.928270458938028857503252420673,
0.928270458938028857503252420673, 2.20207593793785912644808211339, 3.43354035348610912285936934605, 5.40667326701574640538640322595, 6.60324184833313543951395616652, 7.86069650027035754840828050147, 9.080455479241642329412342234498, 9.570910591085020315277297599625, 10.89938240932632903668083568738, 11.95137956321721932132607890787