| L(s) = 1 | + (35.0 + 35.0i)7-s − 341. i·11-s + (−453. + 453. i)13-s + (1.12e3 − 1.12e3i)17-s + 1.02e3i·19-s + (1.04e3 + 1.04e3i)23-s − 532.·29-s − 5.49e3·31-s + (989. + 989. i)37-s + 1.94e3i·41-s + (6.72e3 − 6.72e3i)43-s + (−4.30e3 + 4.30e3i)47-s − 1.43e4i·49-s + (2.12e3 + 2.12e3i)53-s + 2.16e4·59-s + ⋯ |
| L(s) = 1 | + (0.270 + 0.270i)7-s − 0.851i·11-s + (−0.743 + 0.743i)13-s + (0.942 − 0.942i)17-s + 0.650i·19-s + (0.411 + 0.411i)23-s − 0.117·29-s − 1.02·31-s + (0.118 + 0.118i)37-s + 0.180i·41-s + (0.554 − 0.554i)43-s + (−0.283 + 0.283i)47-s − 0.853i·49-s + (0.104 + 0.104i)53-s + 0.810·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.103727623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.103727623\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-35.0 - 35.0i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 + 341. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (453. - 453. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (-1.12e3 + 1.12e3i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 - 1.02e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (-1.04e3 - 1.04e3i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 + 532.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.49e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-989. - 989. i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.94e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-6.72e3 + 6.72e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (4.30e3 - 4.30e3i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (-2.12e3 - 2.12e3i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 - 2.16e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.46e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (1.60e4 + 1.60e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 + 1.63e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (1.37e4 - 1.37e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 - 4.42e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (-2.61e3 - 2.61e3i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 + 7.92e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (6.74e3 + 6.74e3i)T + 8.58e9iT^{2} \) |
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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
−9.365043828729819965033835386500, −8.543678317403423971506471830908, −7.62977692562736053775323267982, −6.89448703128588884340473886781, −5.71652001697039515998162896981, −5.11926696702928665062299820707, −3.91316623781178477171483613583, −2.93721610423616729592878373556, −1.81093212673684198698906979066, −0.61163075286542022405914069961,
0.68722418498050220905066901395, 1.85914412702793854220974718127, 2.95335015295113581802430073921, 4.08859188183455571813189601794, 4.99972643140143524568944496859, 5.83367208343917520702634542159, 7.03665269714775542260475306017, 7.60963142859422892192877419638, 8.488734803578076404151943869621, 9.502257898273937497076545330850
