| L(s) = 1 | + 4.77·2-s − 6.98·3-s + 14.8·4-s − 18.7·5-s − 33.3·6-s + 19.3·7-s + 32.5·8-s + 21.7·9-s − 89.7·10-s − 103.·12-s − 13·13-s + 92.4·14-s + 131.·15-s + 36.8·16-s − 60.0·17-s + 104.·18-s − 113.·19-s − 278.·20-s − 135.·21-s + 126.·23-s − 227.·24-s + 227.·25-s − 62.0·26-s + 36.3·27-s + 286.·28-s + 23.3·29-s + 626.·30-s + ⋯ |
| L(s) = 1 | + 1.68·2-s − 1.34·3-s + 1.85·4-s − 1.67·5-s − 2.26·6-s + 1.04·7-s + 1.43·8-s + 0.807·9-s − 2.83·10-s − 2.48·12-s − 0.277·13-s + 1.76·14-s + 2.25·15-s + 0.576·16-s − 0.856·17-s + 1.36·18-s − 1.36·19-s − 3.11·20-s − 1.40·21-s + 1.14·23-s − 1.93·24-s + 1.82·25-s − 0.468·26-s + 0.259·27-s + 1.93·28-s + 0.149·29-s + 3.81·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.155845239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.155845239\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 - 4.77T + 8T^{2} \) |
| 3 | \( 1 + 6.98T + 27T^{2} \) |
| 5 | \( 1 + 18.7T + 125T^{2} \) |
| 7 | \( 1 - 19.3T + 343T^{2} \) |
| 17 | \( 1 + 60.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 113.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 126.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 23.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 195.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 298.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 409.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 3.52T + 7.95e4T^{2} \) |
| 47 | \( 1 - 106.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 38.5T + 1.48e5T^{2} \) |
| 59 | \( 1 - 752.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 139.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 684.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 577.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 224.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 629.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 161.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.52e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 988.T + 9.12e5T^{2} \) |
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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
−8.809459627400251488642062661160, −7.968992668263766113539973728918, −6.99453738949176742544235006283, −6.56510482418319778501489642663, −5.42732184634687252251708340647, −4.79233920350970129486412745012, −4.35888715753526766784919324170, −3.51311648999012829819069933495, −2.18080769934364808843796681521, −0.57052829224215391898306476596,
0.57052829224215391898306476596, 2.18080769934364808843796681521, 3.51311648999012829819069933495, 4.35888715753526766784919324170, 4.79233920350970129486412745012, 5.42732184634687252251708340647, 6.56510482418319778501489642663, 6.99453738949176742544235006283, 7.968992668263766113539973728918, 8.809459627400251488642062661160
