L(s) = 1 | + 2-s + 3.24·3-s + 4-s − 5-s + 3.24·6-s − 7-s + 8-s + 7.51·9-s − 10-s + 3.24·12-s + 2.14·13-s − 14-s − 3.24·15-s + 16-s + 1.43·17-s + 7.51·18-s − 0.633·19-s − 20-s − 3.24·21-s − 5.61·23-s + 3.24·24-s + 25-s + 2.14·26-s + 14.6·27-s − 28-s − 1.19·29-s − 3.24·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.87·3-s + 0.5·4-s − 0.447·5-s + 1.32·6-s − 0.377·7-s + 0.353·8-s + 2.50·9-s − 0.316·10-s + 0.936·12-s + 0.594·13-s − 0.267·14-s − 0.837·15-s + 0.250·16-s + 0.349·17-s + 1.77·18-s − 0.145·19-s − 0.223·20-s − 0.707·21-s − 1.17·23-s + 0.662·24-s + 0.200·25-s + 0.420·26-s + 2.82·27-s − 0.188·28-s − 0.222·29-s − 0.592·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.750192342\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.750192342\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 3.24T + 3T^{2} \) |
| 13 | \( 1 - 2.14T + 13T^{2} \) |
| 17 | \( 1 - 1.43T + 17T^{2} \) |
| 19 | \( 1 + 0.633T + 19T^{2} \) |
| 23 | \( 1 + 5.61T + 23T^{2} \) |
| 29 | \( 1 + 1.19T + 29T^{2} \) |
| 31 | \( 1 - 6.95T + 31T^{2} \) |
| 37 | \( 1 - 4.80T + 37T^{2} \) |
| 41 | \( 1 + 8.93T + 41T^{2} \) |
| 43 | \( 1 - 9.70T + 43T^{2} \) |
| 47 | \( 1 - 8.29T + 47T^{2} \) |
| 53 | \( 1 - 14.1T + 53T^{2} \) |
| 59 | \( 1 + 1.19T + 59T^{2} \) |
| 61 | \( 1 + 4.19T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 5.71T + 71T^{2} \) |
| 73 | \( 1 - 6.43T + 73T^{2} \) |
| 79 | \( 1 + 7.69T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 2.96T + 97T^{2} \) |
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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
−7.73031826076976371811516951027, −7.33370349803931097474681230259, −6.48172558893643160548671484372, −5.77947592255442018270122925932, −4.59322499709738341222269943237, −4.02628357656133709036580418006, −3.53469327639876763298889895799, −2.77887181567991881106630115922, −2.17385134441837913622240390129, −1.10290541643892074970516549485,
1.10290541643892074970516549485, 2.17385134441837913622240390129, 2.77887181567991881106630115922, 3.53469327639876763298889895799, 4.02628357656133709036580418006, 4.59322499709738341222269943237, 5.77947592255442018270122925932, 6.48172558893643160548671484372, 7.33370349803931097474681230259, 7.73031826076976371811516951027