| L(s) = 1 | + 2.49·2-s + 4.20·4-s + 2.20·5-s + 5.49·8-s + 5.49·10-s + 11-s + 3.28·13-s + 5.26·16-s + 1.49·17-s + 6.91·19-s + 9.26·20-s + 2.49·22-s − 6.49·23-s − 0.140·25-s + 8.18·26-s + 1.64·29-s + 2.35·31-s + 2.14·32-s + 3.71·34-s − 5.55·37-s + 17.2·38-s + 12.1·40-s − 11.2·41-s + 5.26·43-s + 4.20·44-s − 16.1·46-s + 1.49·47-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 2.10·4-s + 0.985·5-s + 1.94·8-s + 1.73·10-s + 0.301·11-s + 0.911·13-s + 1.31·16-s + 0.361·17-s + 1.58·19-s + 2.07·20-s + 0.531·22-s − 1.35·23-s − 0.0281·25-s + 1.60·26-s + 0.306·29-s + 0.422·31-s + 0.378·32-s + 0.636·34-s − 0.913·37-s + 2.79·38-s + 1.91·40-s − 1.75·41-s + 0.803·43-s + 0.633·44-s − 2.38·46-s + 0.217·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.829090431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.829090431\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.49T + 2T^{2} \) |
| 5 | \( 1 - 2.20T + 5T^{2} \) |
| 13 | \( 1 - 3.28T + 13T^{2} \) |
| 17 | \( 1 - 1.49T + 17T^{2} \) |
| 19 | \( 1 - 6.91T + 19T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 - 1.64T + 29T^{2} \) |
| 31 | \( 1 - 2.35T + 31T^{2} \) |
| 37 | \( 1 + 5.55T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 5.26T + 43T^{2} \) |
| 47 | \( 1 - 1.49T + 47T^{2} \) |
| 53 | \( 1 + 0.304T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 4.57T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 8.56T + 73T^{2} \) |
| 79 | \( 1 - 4.63T + 79T^{2} \) |
| 83 | \( 1 + 1.93T + 83T^{2} \) |
| 89 | \( 1 - 3.20T + 89T^{2} \) |
| 97 | \( 1 + 1.85T + 97T^{2} \) |
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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
−8.061758892873817926833938729953, −7.21279011455746323061411562326, −6.44874384327187820095265962024, −5.88676750696864865766974140166, −5.42552200247530533444261274832, −4.64187369408277660756196863364, −3.70298252841855573449015592285, −3.19086913292068307772041663155, −2.14976817219960933533198551941, −1.36907692918477724386975595162,
1.36907692918477724386975595162, 2.14976817219960933533198551941, 3.19086913292068307772041663155, 3.70298252841855573449015592285, 4.64187369408277660756196863364, 5.42552200247530533444261274832, 5.88676750696864865766974140166, 6.44874384327187820095265962024, 7.21279011455746323061411562326, 8.061758892873817926833938729953
