| L(s) = 1 | − 1.15·2-s + 0.330·4-s + 1.36·5-s + 0.772·8-s + 9-s − 1.57·10-s − 1.22·16-s + 1.70·17-s − 1.15·18-s + 0.450·20-s + 0.863·25-s + 0.635·32-s − 1.97·34-s + 0.330·36-s + 0.920·37-s + 1.05·40-s − 1.55·43-s + 1.36·45-s − 1.83·47-s + 49-s − 0.995·50-s − 1.83·61-s + 0.487·64-s − 1.98·67-s + 0.564·68-s + 0.406·71-s + 0.772·72-s + ⋯ |
| L(s) = 1 | − 1.15·2-s + 0.330·4-s + 1.36·5-s + 0.772·8-s + 9-s − 1.57·10-s − 1.22·16-s + 1.70·17-s − 1.15·18-s + 0.450·20-s + 0.863·25-s + 0.635·32-s − 1.97·34-s + 0.330·36-s + 0.920·37-s + 1.05·40-s − 1.55·43-s + 1.36·45-s − 1.83·47-s + 49-s − 0.995·50-s − 1.83·61-s + 0.487·64-s − 1.98·67-s + 0.564·68-s + 0.406·71-s + 0.772·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9473907758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9473907758\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2671 | \( 1+O(T) \) |
| good | 2 | \( 1 + 1.15T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 1.36T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.70T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 0.920T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.55T + T^{2} \) |
| 47 | \( 1 + 1.83T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.83T + T^{2} \) |
| 67 | \( 1 + 1.98T + T^{2} \) |
| 71 | \( 1 - 0.406T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 0.920T + T^{2} \) |
| 89 | \( 1 + 1.55T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.365523251486842786630845551547, −8.315767279779447706914871229112, −7.68978459388455270133556524252, −6.92552007839663194133285865090, −6.08485252215016575194609199379, −5.23812652114532779496633216765, −4.40299893439356930615790002755, −3.13572154741452605402684104721, −1.83280758135487650416813164397, −1.24452081939499910252554918655,
1.24452081939499910252554918655, 1.83280758135487650416813164397, 3.13572154741452605402684104721, 4.40299893439356930615790002755, 5.23812652114532779496633216765, 6.08485252215016575194609199379, 6.92552007839663194133285865090, 7.68978459388455270133556524252, 8.315767279779447706914871229112, 9.365523251486842786630845551547
