L(s) = 1 | + 2-s + 3-s + 4-s − 1.52·5-s + 6-s + 4.59·7-s + 8-s + 9-s − 1.52·10-s + 5.46·11-s + 12-s − 13-s + 4.59·14-s − 1.52·15-s + 16-s − 3.99·17-s + 18-s − 4.36·19-s − 1.52·20-s + 4.59·21-s + 5.46·22-s − 0.656·23-s + 24-s − 2.67·25-s − 26-s + 27-s + 4.59·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.682·5-s + 0.408·6-s + 1.73·7-s + 0.353·8-s + 0.333·9-s − 0.482·10-s + 1.64·11-s + 0.288·12-s − 0.277·13-s + 1.22·14-s − 0.394·15-s + 0.250·16-s − 0.969·17-s + 0.235·18-s − 1.00·19-s − 0.341·20-s + 1.00·21-s + 1.16·22-s − 0.136·23-s + 0.204·24-s − 0.534·25-s − 0.196·26-s + 0.192·27-s + 0.868·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.110618735\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.110618735\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 1.52T + 5T^{2} \) |
| 7 | \( 1 - 4.59T + 7T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 17 | \( 1 + 3.99T + 17T^{2} \) |
| 19 | \( 1 + 4.36T + 19T^{2} \) |
| 23 | \( 1 + 0.656T + 23T^{2} \) |
| 29 | \( 1 - 5.77T + 29T^{2} \) |
| 31 | \( 1 - 7.53T + 31T^{2} \) |
| 37 | \( 1 - 5.61T + 37T^{2} \) |
| 41 | \( 1 - 3.17T + 41T^{2} \) |
| 43 | \( 1 + 8.63T + 43T^{2} \) |
| 47 | \( 1 - 4.76T + 47T^{2} \) |
| 53 | \( 1 - 5.30T + 53T^{2} \) |
| 59 | \( 1 - 1.34T + 59T^{2} \) |
| 61 | \( 1 - 8.91T + 61T^{2} \) |
| 67 | \( 1 - 2.64T + 67T^{2} \) |
| 71 | \( 1 - 2.04T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 1.80T + 79T^{2} \) |
| 83 | \( 1 + 0.725T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 2.62T + 97T^{2} \) |
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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
−7.987458355326000032480008165688, −7.01970398860267834950514716229, −6.60841805766173156355493222235, −5.68146966208172782903690039446, −4.52390605527119467288071981247, −4.42593634156284868607965585021, −3.82672237656628248542059691214, −2.63659191316849308561742514803, −1.92646891694993146575789618779, −1.05487874709382298750313115394,
1.05487874709382298750313115394, 1.92646891694993146575789618779, 2.63659191316849308561742514803, 3.82672237656628248542059691214, 4.42593634156284868607965585021, 4.52390605527119467288071981247, 5.68146966208172782903690039446, 6.60841805766173156355493222235, 7.01970398860267834950514716229, 7.987458355326000032480008165688