| L(s) = 1 | + (−1.85 + 15.4i)3-s − 109. i·5-s − 49i·7-s + (−236. − 57.3i)9-s + 168.·11-s + 624.·13-s + (1.69e3 + 203. i)15-s − 701. i·17-s + 1.07e3i·19-s + (758. + 90.7i)21-s + 3.35e3·23-s − 8.88e3·25-s + (1.32e3 − 3.54e3i)27-s − 3.91e3i·29-s − 1.22e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.118 + 0.992i)3-s − 1.96i·5-s − 0.377i·7-s + (−0.971 − 0.235i)9-s + 0.421·11-s + 1.02·13-s + (1.94 + 0.233i)15-s − 0.588i·17-s + 0.680i·19-s + (0.375 + 0.0449i)21-s + 1.32·23-s − 2.84·25-s + (0.349 − 0.936i)27-s − 0.865i·29-s − 0.228i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.288395939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.288395939\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.85 - 15.4i)T \) |
| 7 | \( 1 + 49iT \) |
| good | 5 | \( 1 + 109. iT - 3.12e3T^{2} \) |
| 11 | \( 1 - 168.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 624.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 701. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.07e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 3.35e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.91e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 1.22e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 8.57e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.16e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 179. iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.17e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.13e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.19e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.13e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.24e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.69e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.49e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.40e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 8.11e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.19e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.04e5T + 8.58e9T^{2} \) |
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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
−10.26244128000996089952379361976, −9.269984969470592962109403534537, −8.818766150050896153639859814871, −7.891296815213285626050460760698, −6.17172635698658407170548108786, −5.18305917231522320752296497011, −4.44690343314810018863479026291, −3.51531685001780985622208746770, −1.42688488137662561737522595569, −0.35110522513355898973730811348,
1.48292069016083475216504060071, 2.71506789985026269792596268121, 3.51927373851079361789741774890, 5.52003684417173816600564180117, 6.67857199536241115130961853319, 6.80881222564359014633586220477, 8.051779686024058567377226836718, 9.060192986369872571702065965850, 10.47760974380255893693362688597, 11.11993591603149946952319075352
