| L(s) = 1 | + (7.53 + 14.1i)2-s + (36.9 + 11.9i)3-s + (−142. + 212. i)4-s + (−624. − 16.5i)5-s + (108. + 611. i)6-s − 802. i·7-s + (−4.07e3 − 404. i)8-s + (−4.08e3 − 2.97e3i)9-s + (−4.47e3 − 8.94e3i)10-s + (1.31e4 + 1.81e4i)11-s + (−7.80e3 + 6.14e3i)12-s + (−1.61e4 − 1.17e4i)13-s + (1.13e4 − 6.05e3i)14-s + (−2.28e4 − 8.10e3i)15-s + (−2.50e4 − 6.05e4i)16-s + (−2.58e4 − 7.96e4i)17-s + ⋯ |
| L(s) = 1 | + (0.471 + 0.882i)2-s + (0.455 + 0.148i)3-s + (−0.556 + 0.831i)4-s + (−0.999 − 0.0264i)5-s + (0.0840 + 0.471i)6-s − 0.334i·7-s + (−0.995 − 0.0988i)8-s + (−0.623 − 0.452i)9-s + (−0.447 − 0.894i)10-s + (0.901 + 1.24i)11-s + (−0.376 + 0.296i)12-s + (−0.565 − 0.410i)13-s + (0.294 − 0.157i)14-s + (−0.451 − 0.160i)15-s + (−0.381 − 0.924i)16-s + (−0.309 − 0.953i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.48533 - 0.151026i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48533 - 0.151026i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-7.53 - 14.1i)T \) |
| 5 | \( 1 + (624. + 16.5i)T \) |
| good | 3 | \( 1 + (-36.9 - 11.9i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 + 802. iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (-1.31e4 - 1.81e4i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (1.61e4 + 1.17e4i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (2.58e4 + 7.96e4i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (-1.35e5 + 4.39e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (-1.52e5 - 2.10e5i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (-3.08e4 + 9.49e4i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (-1.36e6 + 4.42e5i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (2.18e6 + 1.59e6i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (2.37e6 + 1.72e6i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 + 7.29e5iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (1.50e6 + 4.87e5i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (6.38e5 - 1.96e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (5.70e6 - 7.85e6i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (-2.23e7 + 1.62e7i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (-1.74e7 + 5.67e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (3.19e7 + 1.03e7i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (3.24e7 - 2.35e7i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (2.86e7 + 9.29e6i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (-6.50e7 + 2.11e7i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (-6.65e7 + 4.83e7i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (-9.40e6 + 2.89e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−12.16266122265794168071011377342, −11.68206466127882326442239970572, −9.681320059346394368170283367745, −8.781003784228039759787601306349, −7.52540596151515312334595875040, −6.90177399993006674761558420238, −5.13000866249364308703822031164, −4.04872608569498895419231510908, −2.97684433147673275353123944101, −0.39166337658374878147254664831,
1.14169809688966937770338972950, 2.77600058139239378430477259288, 3.68593225533460909042366192042, 5.04844128182721038136661268716, 6.50350863141697179237117126237, 8.293832570771967484559408128176, 8.898889393976192159581144917751, 10.44387075143230366457413985461, 11.56200471205281825854084831029, 12.00211098820416539205611630512
