| L(s) = 1 | + (−13.8 + 13.8i)3-s + (−156. − 156. i)7-s − 140. i·9-s + 402. i·11-s + (703. + 703. i)13-s + (−899. + 899. i)17-s + 507.·19-s + 4.32e3·21-s + (−858. + 858. i)23-s + (−1.41e3 − 1.41e3i)27-s + 6.73e3i·29-s + 2.29e3i·31-s + (−5.58e3 − 5.58e3i)33-s + (−6.09e3 + 6.09e3i)37-s − 1.94e4·39-s + ⋯ |
| L(s) = 1 | + (−0.888 + 0.888i)3-s + (−1.20 − 1.20i)7-s − 0.578i·9-s + 1.00i·11-s + (1.15 + 1.15i)13-s + (−0.754 + 0.754i)17-s + 0.322·19-s + 2.14·21-s + (−0.338 + 0.338i)23-s + (−0.374 − 0.374i)27-s + 1.48i·29-s + 0.428i·31-s + (−0.891 − 0.891i)33-s + (−0.731 + 0.731i)37-s − 2.05·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.03751881584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03751881584\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (13.8 - 13.8i)T - 243iT^{2} \) |
| 7 | \( 1 + (156. + 156. i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 - 402. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-703. - 703. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (899. - 899. i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 - 507.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (858. - 858. i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 - 6.73e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 2.29e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (6.09e3 - 6.09e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 + 4.40e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-5.36e3 + 5.36e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (2.02e4 + 2.02e4i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (1.01e4 + 1.01e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 - 3.25e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.24e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (1.67e3 + 1.67e3i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 - 2.52e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.53e4 - 2.53e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 - 4.18e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + (4.93e4 - 4.93e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 1.33e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-1.00e5 + 1.00e5i)T - 8.58e9iT^{2} \) |
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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
−10.94033957769077723321846397505, −10.22436458198359194208714888599, −9.675406581424207246123068839198, −8.567312008479027179998314203947, −6.92723497526062135938768242795, −6.59838667443575085687629996800, −5.26433636746673888959814287101, −4.17096905417551397530549897693, −3.61426263652757484828701034030, −1.58054075951549625314732690250,
0.01419651814593453037423308907, 0.823046852885932857977054382431, 2.48963166946978419209480933854, 3.52525513783496350283117816427, 5.41750229574899362129016516643, 6.07453804845036025846961304684, 6.54512831153899154759724559812, 7.911501759486994216541618078123, 8.839955266974942674508988476164, 9.759329772090590715089321694254
