| L(s) = 1 | + (−14.6 + 10.6i)2-s + (62.2 − 191. i)4-s + (−425. − 309. i)5-s + (−184. + 568. i)7-s + (411. + 1.26e3i)8-s + 9.54e3·10-s + (2.67e3 + 3.51e3i)11-s + (6.64e3 − 4.83e3i)13-s + (−3.35e3 − 1.03e4i)14-s + (1.28e3 + 934. i)16-s + (1.85e4 + 1.34e4i)17-s + (−1.20e4 − 3.69e4i)19-s + (−8.56e4 + 6.22e4i)20-s + (−7.67e4 − 2.30e4i)22-s + 1.44e4·23-s + ⋯ |
| L(s) = 1 | + (−1.29 + 0.942i)2-s + (0.486 − 1.49i)4-s + (−1.52 − 1.10i)5-s + (−0.203 + 0.626i)7-s + (0.284 + 0.875i)8-s + 3.01·10-s + (0.605 + 0.795i)11-s + (0.839 − 0.609i)13-s + (−0.326 − 1.00i)14-s + (0.0785 + 0.0570i)16-s + (0.915 + 0.664i)17-s + (−0.401 − 1.23i)19-s + (−2.39 + 1.74i)20-s + (−1.53 − 0.462i)22-s + 0.247·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0508019 - 0.0999392i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0508019 - 0.0999392i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-2.67e3 - 3.51e3i)T \) |
| good | 2 | \( 1 + (14.6 - 10.6i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (425. + 309. i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (184. - 568. i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (-6.64e3 + 4.83e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-1.85e4 - 1.34e4i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (1.20e4 + 3.69e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 - 1.44e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (-1.26e4 + 3.89e4i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (1.00e5 - 7.33e4i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (-1.33e5 + 4.11e5i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (2.39e4 + 7.36e4i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 + 8.03e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (2.07e5 + 6.39e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-8.94e4 + 6.49e4i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (2.77e5 - 8.54e5i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (1.11e6 + 8.06e5i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 - 1.15e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-2.02e6 - 1.46e6i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (5.27e5 - 1.62e6i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (-3.74e6 + 2.72e6i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-3.66e6 - 2.66e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 + 9.62e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (1.41e7 - 1.02e7i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−12.11988520433695445397931011009, −10.88400042900664612173818488918, −9.394096713287023570597893345104, −8.644531735993487764867025023692, −7.928894666909317488557358230692, −6.85462193748115924743199492138, −5.36600594779536433312082722040, −3.82452236840402255704817872897, −1.19288412280949062367664741357, −0.07237398794188211527342554050,
1.18439899855581491700695204258, 3.15110957704510632479232114855, 3.81549670748895848332764162103, 6.57384541077434198928470435957, 7.71032647054986209479231152526, 8.466984148796047558177412124293, 9.858543361837322382765428890454, 10.84892173688842105664218137596, 11.43584581907513592703298061003, 12.18059019731347568874686368680
