| L(s) = 1 | − 2-s + 3-s + 4-s − 3·5-s − 6-s − 7-s − 8-s − 2·9-s + 3·10-s + 6·11-s + 12-s + 13-s + 14-s − 3·15-s + 16-s − 3·17-s + 2·18-s + 2·19-s − 3·20-s − 21-s − 6·22-s − 24-s + 4·25-s − 26-s − 5·27-s − 28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.948·10-s + 1.80·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.727·17-s + 0.471·18-s + 0.458·19-s − 0.670·20-s − 0.218·21-s − 1.27·22-s − 0.204·24-s + 4/5·25-s − 0.196·26-s − 0.962·27-s − 0.188·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5155766512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5155766512\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−17.51877200404443387419304773878, −16.32893427071430805415502474938, −15.26212233610873072978362349109, −14.10812305484994740358213704547, −12.10810868275673738373780840946, −11.21486999262721520146961750481, −9.252086119982469784390910809299, −8.289125051595985253760732717123, −6.76423132327857768444846951818, −3.64028761626013442697591583469,
3.64028761626013442697591583469, 6.76423132327857768444846951818, 8.289125051595985253760732717123, 9.252086119982469784390910809299, 11.21486999262721520146961750481, 12.10810868275673738373780840946, 14.10812305484994740358213704547, 15.26212233610873072978362349109, 16.32893427071430805415502474938, 17.51877200404443387419304773878
