| L(s) = 1 | + (−1.39 + 1.01i)2-s + (−38.6 + 118. i)4-s + (60.6 + 44.0i)5-s + (164. − 506. i)7-s + (−134. − 415. i)8-s − 129.·10-s + (−4.39e3 + 446. i)11-s + (782. − 568. i)13-s + (283. + 873. i)14-s + (−1.23e4 − 8.96e3i)16-s + (−2.06e4 − 1.49e4i)17-s + (1.27e4 + 3.91e4i)19-s + (−7.58e3 + 5.51e3i)20-s + (5.67e3 − 5.07e3i)22-s + 1.07e5·23-s + ⋯ |
| L(s) = 1 | + (−0.123 + 0.0895i)2-s + (−0.301 + 0.928i)4-s + (0.217 + 0.157i)5-s + (0.181 − 0.557i)7-s + (−0.0931 − 0.286i)8-s − 0.0409·10-s + (−0.994 + 0.101i)11-s + (0.0987 − 0.0717i)13-s + (0.0276 + 0.0850i)14-s + (−0.753 − 0.547i)16-s + (−1.01 − 0.740i)17-s + (0.425 + 1.31i)19-s + (−0.212 + 0.154i)20-s + (0.113 − 0.101i)22-s + 1.84·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.841826 - 0.563064i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.841826 - 0.563064i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (4.39e3 - 446. i)T \) |
| good | 2 | \( 1 + (1.39 - 1.01i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (-60.6 - 44.0i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (-164. + 506. i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (-782. + 568. i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (2.06e4 + 1.49e4i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (-1.27e4 - 3.91e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 - 1.07e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + (-6.05e4 + 1.86e5i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (1.73e5 - 1.26e5i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (-8.24e4 + 2.53e5i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (2.02e5 + 6.21e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 - 1.94e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + (8.88e4 + 2.73e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-1.34e6 + 9.78e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (-6.75e5 + 2.07e6i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (1.30e6 + 9.48e5i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 - 1.85e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-2.86e5 - 2.08e5i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (1.05e6 - 3.23e6i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (-3.34e6 + 2.43e6i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (4.31e6 + 3.13e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 + 4.39e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + (9.82e6 - 7.13e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−12.44477578198879251482858629839, −11.22272055062418005716446994936, −10.13563501924663223612879488669, −8.890878283690536810018905813782, −7.80902121862783907648156122028, −6.91204885569974192440787507876, −5.16804202350616287501339976306, −3.84572405278625605380752981430, −2.46135887409689599094068219429, −0.35429227381564002530472762956,
1.22448793673431614582551054020, 2.66554467920980916590393833060, 4.78657773546509933463310998055, 5.58306530434668244969141762333, 6.97148152314626658152437565732, 8.660048327967299901954250700856, 9.320598380002435018351360027695, 10.66766864291301564757861609380, 11.30974422380634839825379117052, 13.00891304886232875877489825989
