| 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
−12.00211098820416539205611630512, −11.56200471205281825854084831029, −10.44387075143230366457413985461, −8.898889393976192159581144917751, −8.293832570771967484559408128176, −6.50350863141697179237117126237, −5.04844128182721038136661268716, −3.68593225533460909042366192042, −2.77600058139239378430477259288, −1.14169809688966937770338972950,
0.39166337658374878147254664831, 2.97684433147673275353123944101, 4.04872608569498895419231510908, 5.13000866249364308703822031164, 6.90177399993006674761558420238, 7.52540596151515312334595875040, 8.781003784228039759787601306349, 9.681320059346394368170283367745, 11.68206466127882326442239970572, 12.16266122265794168071011377342
