L(s) = 1 | + 2-s + 0.949·3-s + 4-s + 3.98·5-s + 0.949·6-s − 0.436·7-s + 8-s − 2.09·9-s + 3.98·10-s + 5.12·11-s + 0.949·12-s + 7.08·13-s − 0.436·14-s + 3.78·15-s + 16-s − 3.09·17-s − 2.09·18-s + 3.41·19-s + 3.98·20-s − 0.414·21-s + 5.12·22-s − 1.25·23-s + 0.949·24-s + 10.8·25-s + 7.08·26-s − 4.84·27-s − 0.436·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.548·3-s + 0.5·4-s + 1.78·5-s + 0.387·6-s − 0.164·7-s + 0.353·8-s − 0.699·9-s + 1.26·10-s + 1.54·11-s + 0.274·12-s + 1.96·13-s − 0.116·14-s + 0.977·15-s + 0.250·16-s − 0.749·17-s − 0.494·18-s + 0.784·19-s + 0.891·20-s − 0.0904·21-s + 1.09·22-s − 0.262·23-s + 0.193·24-s + 2.17·25-s + 1.39·26-s − 0.931·27-s − 0.0824·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.644697060\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.644697060\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2003 | \( 1 + T \) |
good | 3 | \( 1 - 0.949T + 3T^{2} \) |
| 5 | \( 1 - 3.98T + 5T^{2} \) |
| 7 | \( 1 + 0.436T + 7T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 - 7.08T + 13T^{2} \) |
| 17 | \( 1 + 3.09T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 + 1.25T + 23T^{2} \) |
| 29 | \( 1 + 5.71T + 29T^{2} \) |
| 31 | \( 1 + 5.45T + 31T^{2} \) |
| 37 | \( 1 + 7.29T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 3.47T + 43T^{2} \) |
| 47 | \( 1 + 8.37T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 0.298T + 71T^{2} \) |
| 73 | \( 1 + 0.271T + 73T^{2} \) |
| 79 | \( 1 + 4.92T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 + 6.47T + 97T^{2} \) |
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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
−8.726047288724074675592259620918, −7.65803486924739174166006228932, −6.55655403037399128811001176151, −6.10181650198914633868220264142, −5.78025981237789760526991977765, −4.70066374598079395333339068493, −3.60080780746036653012187285381, −3.15165780390321476958038715385, −1.89420310401037806332873808148, −1.46456483337051585370539415343,
1.46456483337051585370539415343, 1.89420310401037806332873808148, 3.15165780390321476958038715385, 3.60080780746036653012187285381, 4.70066374598079395333339068493, 5.78025981237789760526991977765, 6.10181650198914633868220264142, 6.55655403037399128811001176151, 7.65803486924739174166006228932, 8.726047288724074675592259620918