| L(s) = 1 | + (4.5 + 7.79i)3-s + (−30.1 + 52.2i)5-s + (84.0 − 98.7i)7-s + (−40.5 + 70.1i)9-s + (3.84 + 6.65i)11-s − 42.5·13-s − 543.·15-s + (−68.9 − 119. i)17-s + (−542. + 940. i)19-s + (1.14e3 + 210. i)21-s + (−1.32e3 + 2.30e3i)23-s + (−258. − 447. i)25-s − 729·27-s − 2.63e3·29-s + (1.28e3 + 2.22e3i)31-s + ⋯ |
| L(s) = 1 | + (0.288 + 0.499i)3-s + (−0.539 + 0.934i)5-s + (0.648 − 0.761i)7-s + (−0.166 + 0.288i)9-s + (0.00957 + 0.0165i)11-s − 0.0698·13-s − 0.623·15-s + (−0.0578 − 0.100i)17-s + (−0.344 + 0.597i)19-s + (0.567 + 0.104i)21-s + (−0.523 + 0.907i)23-s + (−0.0827 − 0.143i)25-s − 0.192·27-s − 0.581·29-s + (0.240 + 0.416i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.394i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4806536140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4806536140\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.5 - 7.79i)T \) |
| 7 | \( 1 + (-84.0 + 98.7i)T \) |
| good | 5 | \( 1 + (30.1 - 52.2i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-3.84 - 6.65i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + 42.5T + 3.71e5T^{2} \) |
| 17 | \( 1 + (68.9 + 119. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (542. - 940. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (1.32e3 - 2.30e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 2.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-1.28e3 - 2.22e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-2.87e3 + 4.97e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 1.81e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.72e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (7.38e3 - 1.27e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (9.39e3 + 1.62e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (3.97e3 + 6.88e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-2.54e4 + 4.41e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.16e4 + 2.02e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 6.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (1.00e4 + 1.73e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (2.35e4 - 4.08e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 4.22e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (1.71e4 - 2.97e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 8.56e4T + 8.58e9T^{2} \) |
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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
−11.10921634926167170133708593866, −10.43061028448201396681984598931, −9.555427117778701797144317914446, −8.242864617905399840189099607576, −7.57160608837738616536507753665, −6.60925407943976738789326316915, −5.18609638112512969696615771791, −4.02241641137441404223065787992, −3.25070302668926721558639257252, −1.73169809051666578493777286427,
0.11574406731437508867417981264, 1.45183825108086929446405328154, 2.63527749397749678606279946996, 4.21137171197185837281931011734, 5.12508188487750105674622721344, 6.30033946988151436022213313877, 7.53618719108239786577833294601, 8.511244087189725581970748113299, 8.799984099599046226882193461489, 10.14613643532913429014479645080
