| 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
−13.81961675213024684509169038605, −12.28033712007313804280316772046, −11.52638639396470955782045350052, −10.37030737500614920941885644904, −9.585075024257450619504082011601, −8.430193193581977599360739493799, −7.27578029705224166021929780337, −5.24191083531772416446369453900, −3.10873874477413709503383907665, −1.23455788606879988094533948817,
1.09455293202096669752893700588, 4.13805431930068928218547986917, 6.26100964793879124436186850889, 7.21659070437691143637360100882, 8.211652203615496922714271724548, 9.547095805530991655375311044254, 10.17153137025392652516386179521, 11.90358716409659349887020844846, 12.90018528202949890489036695616, 14.57903321692530276452205901829
