| L(s) = 1 | − 2.49·2-s + 1.36·3-s + 4.21·4-s − 1.81·5-s − 3.39·6-s + 0.512·7-s − 5.50·8-s − 1.14·9-s + 4.52·10-s − 3.00·11-s + 5.72·12-s − 2.35·13-s − 1.27·14-s − 2.46·15-s + 5.30·16-s + 3.40·17-s + 2.86·18-s − 3.31·19-s − 7.63·20-s + 0.697·21-s + 7.49·22-s − 4.63·23-s − 7.49·24-s − 1.70·25-s + 5.87·26-s − 5.64·27-s + 2.15·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 0.785·3-s + 2.10·4-s − 0.811·5-s − 1.38·6-s + 0.193·7-s − 1.94·8-s − 0.382·9-s + 1.42·10-s − 0.907·11-s + 1.65·12-s − 0.654·13-s − 0.341·14-s − 0.637·15-s + 1.32·16-s + 0.824·17-s + 0.674·18-s − 0.760·19-s − 1.70·20-s + 0.152·21-s + 1.59·22-s − 0.965·23-s − 1.52·24-s − 0.341·25-s + 1.15·26-s − 1.08·27-s + 0.407·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4174199700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4174199700\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5077 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 3 | \( 1 - 1.36T + 3T^{2} \) |
| 5 | \( 1 + 1.81T + 5T^{2} \) |
| 7 | \( 1 - 0.512T + 7T^{2} \) |
| 11 | \( 1 + 3.00T + 11T^{2} \) |
| 13 | \( 1 + 2.35T + 13T^{2} \) |
| 17 | \( 1 - 3.40T + 17T^{2} \) |
| 19 | \( 1 + 3.31T + 19T^{2} \) |
| 23 | \( 1 + 4.63T + 23T^{2} \) |
| 29 | \( 1 - 4.84T + 29T^{2} \) |
| 31 | \( 1 + 7.08T + 31T^{2} \) |
| 37 | \( 1 + 3.49T + 37T^{2} \) |
| 41 | \( 1 - 9.86T + 41T^{2} \) |
| 43 | \( 1 - 2.36T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 6.73T + 53T^{2} \) |
| 59 | \( 1 + 3.76T + 59T^{2} \) |
| 61 | \( 1 + 5.85T + 61T^{2} \) |
| 67 | \( 1 + 7.69T + 67T^{2} \) |
| 71 | \( 1 - 0.317T + 71T^{2} \) |
| 73 | \( 1 - 1.09T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 0.399T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−8.031684340445378378972828713652, −7.82512506649653026164140233651, −7.46784205343188633280610535504, −6.36353069248634546978177929534, −5.53950666402215488991546037273, −4.36904895488981269471798671628, −3.34643766192976789360006374368, −2.54168597833795212303824155424, −1.86097652113631610136243292313, −0.41709347599443254848941830598,
0.41709347599443254848941830598, 1.86097652113631610136243292313, 2.54168597833795212303824155424, 3.34643766192976789360006374368, 4.36904895488981269471798671628, 5.53950666402215488991546037273, 6.36353069248634546978177929534, 7.46784205343188633280610535504, 7.82512506649653026164140233651, 8.031684340445378378972828713652
