L(s) = 1 | + 2-s + 1.17·3-s + 4-s − 5-s + 1.17·6-s + 1.86·7-s + 8-s − 1.61·9-s − 10-s − 1.45·11-s + 1.17·12-s − 2.35·13-s + 1.86·14-s − 1.17·15-s + 16-s + 0.777·17-s − 1.61·18-s − 7.57·19-s − 20-s + 2.19·21-s − 1.45·22-s − 1.98·23-s + 1.17·24-s + 25-s − 2.35·26-s − 5.43·27-s + 1.86·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.679·3-s + 0.5·4-s − 0.447·5-s + 0.480·6-s + 0.703·7-s + 0.353·8-s − 0.537·9-s − 0.316·10-s − 0.440·11-s + 0.339·12-s − 0.652·13-s + 0.497·14-s − 0.304·15-s + 0.250·16-s + 0.188·17-s − 0.380·18-s − 1.73·19-s − 0.223·20-s + 0.478·21-s − 0.311·22-s − 0.413·23-s + 0.240·24-s + 0.200·25-s − 0.461·26-s − 1.04·27-s + 0.351·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 601 | \( 1 - T \) |
good | 3 | \( 1 - 1.17T + 3T^{2} \) |
| 7 | \( 1 - 1.86T + 7T^{2} \) |
| 11 | \( 1 + 1.45T + 11T^{2} \) |
| 13 | \( 1 + 2.35T + 13T^{2} \) |
| 17 | \( 1 - 0.777T + 17T^{2} \) |
| 19 | \( 1 + 7.57T + 19T^{2} \) |
| 23 | \( 1 + 1.98T + 23T^{2} \) |
| 29 | \( 1 - 8.14T + 29T^{2} \) |
| 31 | \( 1 + 0.725T + 31T^{2} \) |
| 37 | \( 1 + 5.70T + 37T^{2} \) |
| 41 | \( 1 + 8.44T + 41T^{2} \) |
| 43 | \( 1 - 3.91T + 43T^{2} \) |
| 47 | \( 1 + 1.60T + 47T^{2} \) |
| 53 | \( 1 + 0.320T + 53T^{2} \) |
| 59 | \( 1 + 9.73T + 59T^{2} \) |
| 61 | \( 1 + 5.49T + 61T^{2} \) |
| 67 | \( 1 - 7.96T + 67T^{2} \) |
| 71 | \( 1 + 1.16T + 71T^{2} \) |
| 73 | \( 1 - 1.72T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 9.39T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 5.98T + 97T^{2} \) |
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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
−7.965633097906057598488371445395, −6.98228191817773280989076973825, −6.32885201036583959395930729162, −5.39220986079146688811225676512, −4.74716927901582537119625778875, −4.09456976474960007459032183362, −3.19150688086797071209975791251, −2.50247235801266915021567056112, −1.72120280037841592124821569776, 0,
1.72120280037841592124821569776, 2.50247235801266915021567056112, 3.19150688086797071209975791251, 4.09456976474960007459032183362, 4.74716927901582537119625778875, 5.39220986079146688811225676512, 6.32885201036583959395930729162, 6.98228191817773280989076973825, 7.965633097906057598488371445395