L(s) = 1 | + 2.59·2-s + 3-s + 4.73·4-s − 2.19·5-s + 2.59·6-s + 7.08·8-s + 9-s − 5.68·10-s + 11-s + 4.73·12-s + 2.95·13-s − 2.19·15-s + 8.92·16-s − 2.59·17-s + 2.59·18-s + 2.35·19-s − 10.3·20-s + 2.59·22-s + 8.92·23-s + 7.08·24-s − 0.195·25-s + 7.66·26-s + 27-s + 4.48·29-s − 5.68·30-s + 1.73·31-s + 8.97·32-s + ⋯ |
L(s) = 1 | + 1.83·2-s + 0.577·3-s + 2.36·4-s − 0.980·5-s + 1.05·6-s + 2.50·8-s + 0.333·9-s − 1.79·10-s + 0.301·11-s + 1.36·12-s + 0.818·13-s − 0.565·15-s + 2.23·16-s − 0.629·17-s + 0.611·18-s + 0.541·19-s − 2.31·20-s + 0.553·22-s + 1.86·23-s + 1.44·24-s − 0.0390·25-s + 1.50·26-s + 0.192·27-s + 0.833·29-s − 1.03·30-s + 0.310·31-s + 1.58·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.850387132\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.850387132\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - 2.59T + 2T^{2} \) |
| 5 | \( 1 + 2.19T + 5T^{2} \) |
| 13 | \( 1 - 2.95T + 13T^{2} \) |
| 17 | \( 1 + 2.59T + 17T^{2} \) |
| 19 | \( 1 - 2.35T + 19T^{2} \) |
| 23 | \( 1 - 8.92T + 23T^{2} \) |
| 29 | \( 1 - 4.48T + 29T^{2} \) |
| 31 | \( 1 - 1.73T + 31T^{2} \) |
| 37 | \( 1 + 4.38T + 37T^{2} \) |
| 41 | \( 1 - 9.68T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 9.17T + 53T^{2} \) |
| 59 | \( 1 + 8.92T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 3.25T + 67T^{2} \) |
| 71 | \( 1 - 0.994T + 71T^{2} \) |
| 73 | \( 1 + 0.294T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 6.26T + 83T^{2} \) |
| 89 | \( 1 - 2.60T + 89T^{2} \) |
| 97 | \( 1 + 7.51T + 97T^{2} \) |
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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
−9.303145430450317948720127972342, −8.386152962411807780490206864923, −7.54285139472778589510240412175, −6.80810490827241025163671797602, −6.11650173624518827151780511815, −4.88548578424811689405684226094, −4.41476566066752217458977125454, −3.37569327522352048045140246715, −3.02373238205682453844855646838, −1.53698908240319652471762740865,
1.53698908240319652471762740865, 3.02373238205682453844855646838, 3.37569327522352048045140246715, 4.41476566066752217458977125454, 4.88548578424811689405684226094, 6.11650173624518827151780511815, 6.80810490827241025163671797602, 7.54285139472778589510240412175, 8.386152962411807780490206864923, 9.303145430450317948720127972342