| L(s) = 1 | + (15.3 + 4.49i)2-s + (215. + 138. i)4-s − 279.·5-s + 820. i·7-s + (2.68e3 + 3.08e3i)8-s + (−4.29e3 − 1.25e3i)10-s − 2.11e4i·11-s − 3.66e4·13-s + (−3.69e3 + 1.26e4i)14-s + (2.74e4 + 5.95e4i)16-s + 1.10e5·17-s − 1.84e5i·19-s + (−6.02e4 − 3.85e4i)20-s + (9.50e4 − 3.24e5i)22-s − 1.81e5i·23-s + ⋯ |
| L(s) = 1 | + (0.959 + 0.280i)2-s + (0.842 + 0.539i)4-s − 0.447·5-s + 0.341i·7-s + (0.656 + 0.754i)8-s + (−0.429 − 0.125i)10-s − 1.44i·11-s − 1.28·13-s + (−0.0960 + 0.328i)14-s + (0.418 + 0.908i)16-s + 1.32·17-s − 1.41i·19-s + (−0.376 − 0.241i)20-s + (0.405 − 1.38i)22-s − 0.650i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 + 0.539i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.842 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.389437888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.389437888\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-15.3 - 4.49i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 279.T \) |
| good | 7 | \( 1 - 820. iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 2.11e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 3.66e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.10e5T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.84e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 1.81e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 2.29e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 3.88e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 1.32e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.03e5T + 7.98e12T^{2} \) |
| 43 | \( 1 - 9.62e5iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 8.76e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 5.77e5T + 6.22e13T^{2} \) |
| 59 | \( 1 - 4.11e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.10e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 3.85e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 2.07e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.70e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 5.62e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 6.22e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 6.04e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 1.31e8T + 7.83e15T^{2} \) |
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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
−11.43463582717383751305968186624, −10.34171741265377169514602128798, −8.816272092765765803303088800057, −7.82434543305517489985104947105, −6.82478769256624288366206809475, −5.64351290719289509791027442569, −4.75321370656399039989552381579, −3.39186311100081611047657914846, −2.52010138560144520678575583432, −0.60130929411634214413318058648,
1.19703306123453533637286194177, 2.45140894703455375454525774213, 3.77090975054320678575558011887, 4.68650923427147748324266064806, 5.76529700670259423019265893637, 7.22329769515278412846491787504, 7.71395322351268920496723387256, 9.794040509580085698282602071852, 10.18912072479747691845727620594, 11.62170357732257411237877727068
