L(s) = 1 | + (10.3 − 11.6i)3-s + 68.3i·5-s − 220. i·7-s + (−28.4 − 241. i)9-s − 127.·11-s − 822.·13-s + (796. + 707. i)15-s − 1.61e3i·17-s + 1.81e3i·19-s + (−2.57e3 − 2.28e3i)21-s + 1.27e3·23-s − 1.54e3·25-s + (−3.10e3 − 2.16e3i)27-s + 7.51e3i·29-s + 8.58e3i·31-s + ⋯ |
L(s) = 1 | + (0.664 − 0.747i)3-s + 1.22i·5-s − 1.70i·7-s + (−0.117 − 0.993i)9-s − 0.316·11-s − 1.34·13-s + (0.913 + 0.812i)15-s − 1.35i·17-s + 1.15i·19-s + (−1.27 − 1.13i)21-s + 0.501·23-s − 0.493·25-s + (−0.820 − 0.572i)27-s + 1.65i·29-s + 1.60i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.06043254474\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06043254474\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-10.3 + 11.6i)T \) |
good | 5 | \( 1 - 68.3iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 220. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 127.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 822.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.61e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.81e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.27e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.51e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 8.58e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 5.30e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.35e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.07e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 4.89e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.95e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.15e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.00e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.46e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.50e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.40e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 1.95e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.97e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.32e5T + 8.58e9T^{2} \) |
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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
−10.22156488980087243156301311953, −9.030797525074770996932386794861, −7.58680655429780654436676908383, −7.25163038580197674290352693028, −6.67304849407669255042049671293, −4.96275024946394186880355321121, −3.50860969091285699561270572570, −2.83111053499037717910034162101, −1.40080925535106334667625520054, −0.01269549160023239933258804334,
1.99060004275429504417903300233, 2.80646098743789170138134131567, 4.43241300028544108436997450068, 5.07464904712453197360955342148, 5.99902151808843130257449376734, 7.81834794713769036917846070957, 8.478510011488114886890481739632, 9.288090933142317598554537670225, 9.726866548491490572488902872877, 11.09188977876623644683589574687