| L(s) = 1 | − 2.72·2-s − 1.52·3-s + 5.43·4-s + 2.93·5-s + 4.15·6-s − 9.36·8-s − 0.681·9-s − 7.99·10-s − 1.46·11-s − 8.27·12-s + 3.01·13-s − 4.46·15-s + 14.6·16-s + 5.83·17-s + 1.85·18-s − 6.08·19-s + 15.9·20-s + 3.98·22-s − 3.59·23-s + 14.2·24-s + 3.60·25-s − 8.20·26-s + 5.60·27-s + 4.02·29-s + 12.1·30-s − 7.40·31-s − 21.2·32-s + ⋯ |
| L(s) = 1 | − 1.92·2-s − 0.879·3-s + 2.71·4-s + 1.31·5-s + 1.69·6-s − 3.31·8-s − 0.227·9-s − 2.52·10-s − 0.440·11-s − 2.38·12-s + 0.834·13-s − 1.15·15-s + 3.66·16-s + 1.41·17-s + 0.438·18-s − 1.39·19-s + 3.56·20-s + 0.850·22-s − 0.749·23-s + 2.91·24-s + 0.720·25-s − 1.60·26-s + 1.07·27-s + 0.747·29-s + 2.22·30-s − 1.33·31-s − 3.76·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 3 | \( 1 + 1.52T + 3T^{2} \) |
| 5 | \( 1 - 2.93T + 5T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 13 | \( 1 - 3.01T + 13T^{2} \) |
| 17 | \( 1 - 5.83T + 17T^{2} \) |
| 19 | \( 1 + 6.08T + 19T^{2} \) |
| 23 | \( 1 + 3.59T + 23T^{2} \) |
| 29 | \( 1 - 4.02T + 29T^{2} \) |
| 31 | \( 1 + 7.40T + 31T^{2} \) |
| 37 | \( 1 + 8.74T + 37T^{2} \) |
| 43 | \( 1 + 8.73T + 43T^{2} \) |
| 47 | \( 1 + 5.39T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 9.68T + 59T^{2} \) |
| 61 | \( 1 + 8.70T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 2.73T + 71T^{2} \) |
| 73 | \( 1 + 1.94T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 6.65T + 83T^{2} \) |
| 89 | \( 1 - 2.46T + 89T^{2} \) |
| 97 | \( 1 + 7.41T + 97T^{2} \) |
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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
−8.685982772689057426609040538072, −8.348367332162724136989137995522, −7.26639066068661411217746414361, −6.37278138648770746729324385570, −5.97048296095056770480883095278, −5.27769406689839311033962100213, −3.30052190791738997535648038683, −2.15093388882201481634125928672, −1.35811667322077932279876778572, 0,
1.35811667322077932279876778572, 2.15093388882201481634125928672, 3.30052190791738997535648038683, 5.27769406689839311033962100213, 5.97048296095056770480883095278, 6.37278138648770746729324385570, 7.26639066068661411217746414361, 8.348367332162724136989137995522, 8.685982772689057426609040538072
