| L(s) = 1 | + (−3.31 + 2.40i)2-s + (−0.224 − 0.689i)3-s + (2.71 − 8.36i)4-s + (−1.39 − 1.01i)5-s + (2.40 + 1.74i)6-s + (2.16 − 6.65i)7-s + (1.00 + 3.08i)8-s + (21.4 − 15.5i)9-s + 7.07·10-s + (28.8 + 22.2i)11-s − 6.37·12-s + (−19.4 + 14.1i)13-s + (8.86 + 27.2i)14-s + (−0.386 + 1.19i)15-s + (46.1 + 33.5i)16-s + (107. + 78.1i)17-s + ⋯ |
| L(s) = 1 | + (−1.17 + 0.851i)2-s + (−0.0431 − 0.132i)3-s + (0.339 − 1.04i)4-s + (−0.124 − 0.0907i)5-s + (0.163 + 0.118i)6-s + (0.116 − 0.359i)7-s + (0.0443 + 0.136i)8-s + (0.793 − 0.576i)9-s + 0.223·10-s + (0.791 + 0.611i)11-s − 0.153·12-s + (−0.415 + 0.301i)13-s + (0.169 + 0.520i)14-s + (−0.00665 + 0.0204i)15-s + (0.720 + 0.523i)16-s + (1.53 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.544i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.839 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.824611 + 0.243989i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.824611 + 0.243989i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.16 + 6.65i)T \) |
| 11 | \( 1 + (-28.8 - 22.2i)T \) |
| good | 2 | \( 1 + (3.31 - 2.40i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (0.224 + 0.689i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (1.39 + 1.01i)T + (38.6 + 118. i)T^{2} \) |
| 13 | \( 1 + (19.4 - 14.1i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-107. - 78.1i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (10.2 + 31.6i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 - 95.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-15.0 + 46.3i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-239. + 174. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-59.0 + 181. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-68.8 - 211. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 302.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-64.7 - 199. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-103. + 75.4i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-119. + 368. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-594. - 432. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + 1.02e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-171. - 124. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (182. - 562. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-292. + 212. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (303. + 220. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 632.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (1.01e3 - 739. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−14.57903321692530276452205901829, −12.90018528202949890489036695616, −11.90358716409659349887020844846, −10.17153137025392652516386179521, −9.547095805530991655375311044254, −8.211652203615496922714271724548, −7.21659070437691143637360100882, −6.26100964793879124436186850889, −4.13805431930068928218547986917, −1.09455293202096669752893700588,
1.23455788606879988094533948817, 3.10873874477413709503383907665, 5.24191083531772416446369453900, 7.27578029705224166021929780337, 8.430193193581977599360739493799, 9.585075024257450619504082011601, 10.37030737500614920941885644904, 11.52638639396470955782045350052, 12.28033712007313804280316772046, 13.81961675213024684509169038605
