L(s) = 1 | − 2-s + 0.757·3-s + 4-s + 1.30·5-s − 0.757·6-s − 7-s − 8-s − 2.42·9-s − 1.30·10-s + 1.45·11-s + 0.757·12-s + 3.28·13-s + 14-s + 0.989·15-s + 16-s + 7.19·17-s + 2.42·18-s + 5.28·19-s + 1.30·20-s − 0.757·21-s − 1.45·22-s − 3.28·23-s − 0.757·24-s − 3.29·25-s − 3.28·26-s − 4.10·27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.437·3-s + 0.5·4-s + 0.584·5-s − 0.309·6-s − 0.377·7-s − 0.353·8-s − 0.808·9-s − 0.413·10-s + 0.437·11-s + 0.218·12-s + 0.910·13-s + 0.267·14-s + 0.255·15-s + 0.250·16-s + 1.74·17-s + 0.571·18-s + 1.21·19-s + 0.292·20-s − 0.165·21-s − 0.309·22-s − 0.684·23-s − 0.154·24-s − 0.658·25-s − 0.643·26-s − 0.790·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.353520621\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.353520621\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 - 0.757T + 3T^{2} \) |
| 5 | \( 1 - 1.30T + 5T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 13 | \( 1 - 3.28T + 13T^{2} \) |
| 17 | \( 1 - 7.19T + 17T^{2} \) |
| 19 | \( 1 - 5.28T + 19T^{2} \) |
| 23 | \( 1 + 3.28T + 23T^{2} \) |
| 29 | \( 1 - 6.46T + 29T^{2} \) |
| 31 | \( 1 - 3.01T + 31T^{2} \) |
| 37 | \( 1 + 3.51T + 37T^{2} \) |
| 43 | \( 1 - 0.525T + 43T^{2} \) |
| 47 | \( 1 - 3.15T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 1.68T + 59T^{2} \) |
| 61 | \( 1 + 0.842T + 61T^{2} \) |
| 67 | \( 1 - 7.88T + 67T^{2} \) |
| 71 | \( 1 + 16.4T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 0.168T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 3.62T + 97T^{2} \) |
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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
−10.35787755626458204761125405828, −9.846616902437775142633327104537, −8.975113858372468198398001570031, −8.246145664499039995386766481044, −7.35629933919345843844553158803, −6.11645933976654703196940785373, −5.56573970648120326831243937314, −3.67814377867018393193813802700, −2.75706114457503498399189516382, −1.23486423036038882779714156875,
1.23486423036038882779714156875, 2.75706114457503498399189516382, 3.67814377867018393193813802700, 5.56573970648120326831243937314, 6.11645933976654703196940785373, 7.35629933919345843844553158803, 8.246145664499039995386766481044, 8.975113858372468198398001570031, 9.846616902437775142633327104537, 10.35787755626458204761125405828