L(s) = 1 | + (−1.24 − 2.15i)5-s + (0.909 − 1.57i)7-s + (0.598 − 1.03i)11-s + (2.83 + 4.90i)13-s + 5.30·17-s + 4.55·19-s + (2.01 + 3.48i)23-s + (−0.589 + 1.02i)25-s + (−3.01 + 5.22i)29-s + (2.81 + 4.87i)31-s − 4.51·35-s − 5.18·37-s + (−4.57 − 7.92i)41-s + (−3.99 + 6.91i)43-s + (−1.39 + 2.41i)47-s + ⋯ |
L(s) = 1 | + (−0.555 − 0.962i)5-s + (0.343 − 0.595i)7-s + (0.180 − 0.312i)11-s + (0.785 + 1.36i)13-s + 1.28·17-s + 1.04·19-s + (0.419 + 0.727i)23-s + (−0.117 + 0.204i)25-s + (−0.559 + 0.969i)29-s + (0.505 + 0.876i)31-s − 0.763·35-s − 0.851·37-s + (−0.714 − 1.23i)41-s + (−0.608 + 1.05i)43-s + (−0.203 + 0.352i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.976519855\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.976519855\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.24 + 2.15i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.909 + 1.57i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.598 + 1.03i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.83 - 4.90i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 5.30T + 17T^{2} \) |
| 19 | \( 1 - 4.55T + 19T^{2} \) |
| 23 | \( 1 + (-2.01 - 3.48i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.01 - 5.22i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.81 - 4.87i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 5.18T + 37T^{2} \) |
| 41 | \( 1 + (4.57 + 7.92i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.99 - 6.91i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.39 - 2.41i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 1.54T + 53T^{2} \) |
| 59 | \( 1 + (1.85 + 3.21i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.01 - 6.95i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.91 - 11.9i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + (-4.36 + 7.56i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.89 + 15.4i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 0.455T + 89T^{2} \) |
| 97 | \( 1 + (1.01 - 1.76i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.653375449800418634795776053033, −7.81455387440005380018655167420, −7.22525014397716615148593916288, −6.38558314516405316052829236306, −5.29807357546618228594056338467, −4.83288009271390404101579219095, −3.80229583914297862443171264506, −3.34250219997410958092820624402, −1.56388421214228641295347905426, −1.00034032458578420553893193170,
0.803370108308543043722156285904, 2.19034544632605589803944484047, 3.28896654965803087428090503460, 3.56521562980352551541204232351, 4.97163646514494392587690371194, 5.56886421590038589381626202642, 6.41398291267031310798044110864, 7.19376081444172662689516594643, 8.080606657224999146753938465750, 8.205301621011246562306663662562