| L(s) = 1 | + (−0.255 + 1.39i)2-s + (1.53 − 2.18i)3-s + (−1.86 − 0.711i)4-s + (0.112 − 1.28i)5-s + (2.64 + 2.68i)6-s + (−0.169 + 0.293i)7-s + (1.46 − 2.41i)8-s + (−1.41 − 3.87i)9-s + (1.75 + 0.483i)10-s + (0.0132 − 0.0492i)11-s + (−4.41 + 2.99i)12-s + (−1.98 − 2.83i)13-s + (−0.364 − 0.310i)14-s + (−2.63 − 2.20i)15-s + (2.98 + 2.65i)16-s + (3.23 + 1.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.180 + 0.983i)2-s + (0.883 − 1.26i)3-s + (−0.934 − 0.355i)4-s + (0.0501 − 0.573i)5-s + (1.08 + 1.09i)6-s + (−0.0639 + 0.110i)7-s + (0.518 − 0.854i)8-s + (−0.470 − 1.29i)9-s + (0.554 + 0.152i)10-s + (0.00398 − 0.0148i)11-s + (−1.27 + 0.865i)12-s + (−0.549 − 0.785i)13-s + (−0.0974 − 0.0829i)14-s + (−0.679 − 0.570i)15-s + (0.747 + 0.664i)16-s + (0.784 + 0.285i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28599 - 0.504056i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28599 - 0.504056i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.255 - 1.39i)T \) |
| 19 | \( 1 + (-0.512 + 4.32i)T \) |
| good | 3 | \( 1 + (-1.53 + 2.18i)T + (-1.02 - 2.81i)T^{2} \) |
| 5 | \( 1 + (-0.112 + 1.28i)T + (-4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (0.169 - 0.293i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.0132 + 0.0492i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (1.98 + 2.83i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-3.23 - 1.17i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (2.55 + 2.14i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.60 + 3.43i)T + (-18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (4.05 - 7.02i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.32 - 4.32i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1.27 - 7.21i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (1.24 + 0.108i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-1.02 - 2.81i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-0.188 - 2.14i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (-5.44 + 2.54i)T + (37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (-1.06 - 12.2i)T + (-60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (-0.697 - 0.325i)T + (43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (8.38 + 9.99i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (4.23 - 0.746i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-1.92 - 10.8i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (3.41 + 12.7i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-2.78 + 15.7i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (4.98 - 13.6i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−12.09375275980014311645694324870, −10.36048839233050922822694382228, −9.257992680048114022005757476626, −8.477254299927765321120520603136, −7.77856416256957797081694810635, −6.99237604786500397397670366450, −5.91711224219944834598460183786, −4.71568284184688781875984790244, −2.93170649001411412417469120100, −1.09407565781067084802897379004,
2.27079594069755330331678250201, 3.44301753309165768459317180043, 4.16631987555305658273269763942, 5.42303799764371528370241687262, 7.34925938623094708066507205372, 8.417073922289291786287132373472, 9.393087276921505581150055218344, 9.928469019384009514650616301795, 10.64229392301200753512104899796, 11.60566683884993107609675358803
