| L(s) = 1 | − 2-s + 0.797·3-s + 4-s − 5-s − 0.797·6-s − 1.54·7-s − 8-s − 2.36·9-s + 10-s − 11-s + 0.797·12-s − 1.10·13-s + 1.54·14-s − 0.797·15-s + 16-s − 3.83·17-s + 2.36·18-s + 7.93·19-s − 20-s − 1.22·21-s + 22-s − 3.10·23-s − 0.797·24-s + 25-s + 1.10·26-s − 4.27·27-s − 1.54·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.460·3-s + 0.5·4-s − 0.447·5-s − 0.325·6-s − 0.582·7-s − 0.353·8-s − 0.788·9-s + 0.316·10-s − 0.301·11-s + 0.230·12-s − 0.307·13-s + 0.412·14-s − 0.205·15-s + 0.250·16-s − 0.931·17-s + 0.557·18-s + 1.82·19-s − 0.223·20-s − 0.268·21-s + 0.213·22-s − 0.648·23-s − 0.162·24-s + 0.200·25-s + 0.217·26-s − 0.823·27-s − 0.291·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8457761335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8457761335\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 - 0.797T + 3T^{2} \) |
| 7 | \( 1 + 1.54T + 7T^{2} \) |
| 13 | \( 1 + 1.10T + 13T^{2} \) |
| 17 | \( 1 + 3.83T + 17T^{2} \) |
| 19 | \( 1 - 7.93T + 19T^{2} \) |
| 23 | \( 1 + 3.10T + 23T^{2} \) |
| 29 | \( 1 + 2.76T + 29T^{2} \) |
| 31 | \( 1 - 3.78T + 31T^{2} \) |
| 37 | \( 1 - 6.96T + 37T^{2} \) |
| 41 | \( 1 + 6.69T + 41T^{2} \) |
| 47 | \( 1 + 2.36T + 47T^{2} \) |
| 53 | \( 1 - 0.0407T + 53T^{2} \) |
| 59 | \( 1 + 14.7T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 9.13T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 + 0.994T + 73T^{2} \) |
| 79 | \( 1 - 2.35T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 - 7.56T + 89T^{2} \) |
| 97 | \( 1 - 6.65T + 97T^{2} \) |
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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
−8.234452016212869448483748430019, −7.76605102462004585558737292294, −7.05326568824467856287659091342, −6.25087749997645793099031508085, −5.49547799505844681299098400311, −4.53795183926067618953898731915, −3.39412665767987732015116094564, −2.92890427442941869543915322173, −1.96166645436441330337968215608, −0.52671822966907940508622333913,
0.52671822966907940508622333913, 1.96166645436441330337968215608, 2.92890427442941869543915322173, 3.39412665767987732015116094564, 4.53795183926067618953898731915, 5.49547799505844681299098400311, 6.25087749997645793099031508085, 7.05326568824467856287659091342, 7.76605102462004585558737292294, 8.234452016212869448483748430019
