| L(s) = 1 | + (0.967 − 1.03i)2-s + (−0.513 − 0.753i)3-s + (−0.126 − 1.99i)4-s + (0.131 + 1.75i)5-s + (−1.27 − 0.199i)6-s + (2.59 + 0.503i)7-s + (−2.18 − 1.80i)8-s + (0.792 − 2.01i)9-s + (1.93 + 1.56i)10-s + (−1.93 − 4.92i)11-s + (−1.43 + 1.12i)12-s + (1.33 − 1.67i)13-s + (3.03 − 2.19i)14-s + (1.25 − 0.999i)15-s + (−3.96 + 0.506i)16-s + (−4.42 + 4.77i)17-s + ⋯ |
| L(s) = 1 | + (0.684 − 0.729i)2-s + (−0.296 − 0.434i)3-s + (−0.0634 − 0.997i)4-s + (0.0587 + 0.784i)5-s + (−0.520 − 0.0814i)6-s + (0.981 + 0.190i)7-s + (−0.771 − 0.636i)8-s + (0.264 − 0.672i)9-s + (0.612 + 0.493i)10-s + (−0.582 − 1.48i)11-s + (−0.415 + 0.323i)12-s + (0.371 − 0.465i)13-s + (0.810 − 0.585i)14-s + (0.323 − 0.258i)15-s + (−0.991 + 0.126i)16-s + (−1.07 + 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.09185 - 1.48613i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09185 - 1.48613i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.967 + 1.03i)T \) |
| 7 | \( 1 + (-2.59 - 0.503i)T \) |
| good | 3 | \( 1 + (0.513 + 0.753i)T + (-1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (-0.131 - 1.75i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (1.93 + 4.92i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (-1.33 + 1.67i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (4.42 - 4.77i)T + (-1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-5.47 + 3.16i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.24 - 3.50i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (5.75 + 1.31i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-0.509 + 0.882i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.61 - 5.21i)T + (-30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (0.546 - 1.13i)T + (-25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (4.10 - 1.97i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-6.05 - 0.912i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (1.80 - 5.86i)T + (-43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-8.59 - 0.644i)T + (58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (0.549 - 0.169i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-0.480 + 0.832i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-14.9 + 3.41i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-2.04 - 13.5i)T + (-69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (6.83 - 3.94i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.376 - 0.300i)T + (18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (3.65 + 1.43i)T + (65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + 4.38iT - 97T^{2} \) |
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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
−11.19683512265997902644740932715, −10.64962162678179385059852099954, −9.361712142892874886260744137891, −8.303281862514118628184704317899, −7.00687557494587436602170643623, −6.02924201856517609767454360543, −5.27148722585936277613780699721, −3.77870992577089889978148630872, −2.75242867575695296564784758619, −1.15388589769400241456360055028,
2.12774758686414304492153701935, 4.13007588751522811352122572828, 4.98088891375331450382129028014, 5.24094534473470764925842182459, 7.02712153574758239247845880921, 7.61331292387207475950605726375, 8.692356086294064889497206336253, 9.612680483837087143183081074146, 10.87774700095815991001443120896, 11.66466836146846445405221491607
