L(s) = 1 | − 0.460·3-s − 1.55·5-s − 7-s − 2.78·9-s + 5.69·11-s − 0.801·13-s + 0.717·15-s + 4.85·17-s − 4.04·19-s + 0.460·21-s + 5.34·23-s − 2.57·25-s + 2.66·27-s − 8.47·29-s − 10.9·31-s − 2.62·33-s + 1.55·35-s + 7.26·37-s + 0.368·39-s + 11.6·41-s − 6.90·43-s + 4.34·45-s + 5.07·47-s + 49-s − 2.23·51-s − 12.2·53-s − 8.87·55-s + ⋯ |
L(s) = 1 | − 0.265·3-s − 0.696·5-s − 0.377·7-s − 0.929·9-s + 1.71·11-s − 0.222·13-s + 0.185·15-s + 1.17·17-s − 0.928·19-s + 0.100·21-s + 1.11·23-s − 0.514·25-s + 0.512·27-s − 1.57·29-s − 1.95·31-s − 0.456·33-s + 0.263·35-s + 1.19·37-s + 0.0590·39-s + 1.81·41-s − 1.05·43-s + 0.647·45-s + 0.739·47-s + 0.142·49-s − 0.312·51-s − 1.67·53-s − 1.19·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.181535264\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.181535264\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + 0.460T + 3T^{2} \) |
| 5 | \( 1 + 1.55T + 5T^{2} \) |
| 11 | \( 1 - 5.69T + 11T^{2} \) |
| 13 | \( 1 + 0.801T + 13T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 19 | \( 1 + 4.04T + 19T^{2} \) |
| 23 | \( 1 - 5.34T + 23T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 - 7.26T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 6.90T + 43T^{2} \) |
| 47 | \( 1 - 5.07T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 1.61T + 59T^{2} \) |
| 61 | \( 1 - 7.21T + 61T^{2} \) |
| 67 | \( 1 + 9.44T + 67T^{2} \) |
| 71 | \( 1 - 1.51T + 71T^{2} \) |
| 73 | \( 1 - 4.21T + 73T^{2} \) |
| 79 | \( 1 + 2.48T + 79T^{2} \) |
| 83 | \( 1 + 6.11T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−8.701672212469891906260782693878, −7.65787123642344664146157865596, −7.20430604300352066810175029720, −6.14779182548566971140447962242, −5.78786139787265994956399696344, −4.67310251137691809159880570626, −3.75499615500193353304887916513, −3.29637369912738882600138029557, −1.94701771383973855384538746851, −0.64117596722209019796145249257,
0.64117596722209019796145249257, 1.94701771383973855384538746851, 3.29637369912738882600138029557, 3.75499615500193353304887916513, 4.67310251137691809159880570626, 5.78786139787265994956399696344, 6.14779182548566971140447962242, 7.20430604300352066810175029720, 7.65787123642344664146157865596, 8.701672212469891906260782693878