| L(s) = 1 | − 2-s − 0.655·3-s + 4-s − 2.51·5-s + 0.655·6-s − 7-s − 8-s − 2.57·9-s + 2.51·10-s + 0.759·11-s − 0.655·12-s − 1.72·13-s + 14-s + 1.64·15-s + 16-s − 4.29·17-s + 2.57·18-s − 4.70·19-s − 2.51·20-s + 0.655·21-s − 0.759·22-s + 6.68·23-s + 0.655·24-s + 1.32·25-s + 1.72·26-s + 3.65·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.378·3-s + 0.5·4-s − 1.12·5-s + 0.267·6-s − 0.377·7-s − 0.353·8-s − 0.856·9-s + 0.795·10-s + 0.228·11-s − 0.189·12-s − 0.477·13-s + 0.267·14-s + 0.425·15-s + 0.250·16-s − 1.04·17-s + 0.605·18-s − 1.07·19-s − 0.562·20-s + 0.143·21-s − 0.161·22-s + 1.39·23-s + 0.133·24-s + 0.265·25-s + 0.337·26-s + 0.702·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 0.655T + 3T^{2} \) |
| 5 | \( 1 + 2.51T + 5T^{2} \) |
| 11 | \( 1 - 0.759T + 11T^{2} \) |
| 13 | \( 1 + 1.72T + 13T^{2} \) |
| 17 | \( 1 + 4.29T + 17T^{2} \) |
| 19 | \( 1 + 4.70T + 19T^{2} \) |
| 23 | \( 1 - 6.68T + 23T^{2} \) |
| 29 | \( 1 - 3.74T + 29T^{2} \) |
| 31 | \( 1 - 8.67T + 31T^{2} \) |
| 37 | \( 1 - 3.60T + 37T^{2} \) |
| 41 | \( 1 - 2.87T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 9.98T + 53T^{2} \) |
| 59 | \( 1 + 4.78T + 59T^{2} \) |
| 61 | \( 1 + 4.02T + 61T^{2} \) |
| 67 | \( 1 + 3.87T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 0.893T + 73T^{2} \) |
| 79 | \( 1 - 2.90T + 79T^{2} \) |
| 83 | \( 1 + 1.48T + 83T^{2} \) |
| 89 | \( 1 + 6.12T + 89T^{2} \) |
| 97 | \( 1 - 0.629T + 97T^{2} \) |
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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
−7.950937481107227048106723370929, −6.89149539788889406842885523536, −6.60635289835655379372243180112, −5.74942316585731421548041542697, −4.69925892618501415131376233389, −4.13898972341746737767298230428, −3.00014126439600847732593842191, −2.44689768672672770211271759948, −0.879113927863667763635689012515, 0,
0.879113927863667763635689012515, 2.44689768672672770211271759948, 3.00014126439600847732593842191, 4.13898972341746737767298230428, 4.69925892618501415131376233389, 5.74942316585731421548041542697, 6.60635289835655379372243180112, 6.89149539788889406842885523536, 7.950937481107227048106723370929
