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
\(L(1)\) |
\(\approx\) |
\(1.276619963\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.276619963\) |
\(L(\frac{3}{2})\) |
|
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
−8.117147115784807517942684484578, −7.85283969458343862861121468416, −7.61617847324713677539828381781, −6.11018859679249593961650596199, −5.55632696413073825654028203226, −4.80890594566278602987496869777, −3.80591576631456449080261665729, −2.95894542267998226215919448862, −2.25106808531878961570675005364, −0.62732469150982519164928931900,
0.62732469150982519164928931900, 2.25106808531878961570675005364, 2.95894542267998226215919448862, 3.80591576631456449080261665729, 4.80890594566278602987496869777, 5.55632696413073825654028203226, 6.11018859679249593961650596199, 7.61617847324713677539828381781, 7.85283969458343862861121468416, 8.117147115784807517942684484578