L(s) = 1 | + (−2.12 + 2.12i)3-s + (−0.706 − 0.706i)5-s − 4.44i·7-s − 8.99i·9-s + (17.7 + 17.7i)11-s + (17.7 − 17.7i)13-s + 2.99·15-s − 105.·17-s + (−40.2 + 40.2i)19-s + (9.42 + 9.42i)21-s − 42.9i·23-s − 124. i·25-s + (19.0 + 19.0i)27-s + (185. − 185. i)29-s + 291.·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (−0.0631 − 0.0631i)5-s − 0.239i·7-s − 0.333i·9-s + (0.485 + 0.485i)11-s + (0.377 − 0.377i)13-s + 0.0516·15-s − 1.50·17-s + (−0.486 + 0.486i)19-s + (0.0978 + 0.0978i)21-s − 0.389i·23-s − 0.992i·25-s + (0.136 + 0.136i)27-s + (1.18 − 1.18i)29-s + 1.68·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.351942966\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.351942966\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.12 - 2.12i)T \) |
good | 5 | \( 1 + (0.706 + 0.706i)T + 125iT^{2} \) |
| 7 | \( 1 + 4.44iT - 343T^{2} \) |
| 11 | \( 1 + (-17.7 - 17.7i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + (-17.7 + 17.7i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + 105.T + 4.91e3T^{2} \) |
| 19 | \( 1 + (40.2 - 40.2i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 + 42.9iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-185. + 185. i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 - 291.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (151. + 151. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 - 60.3iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (120. + 120. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 - 500.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (192. + 192. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (-500. - 500. i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (-166. + 166. i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 + (-575. + 575. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 + 457. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 1.08e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 544.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (48.6 - 48.6i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 - 43.6iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 690.T + 9.12e5T^{2} \) |
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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
−10.63176123108243624000717213183, −10.12771270903553454012410365122, −8.947767776207527817246175202202, −8.155397314603452779981096673058, −6.77401507540302414905066815002, −6.11659456556549885351829514358, −4.67436195273425814353719700768, −4.01920128657025663166255076672, −2.35684463954149462662430305478, −0.56896733034769752911647824158,
1.13391508816204614964635711620, 2.60818976807497724483600534059, 4.10311749672839333156181751114, 5.24414931315363161535495914882, 6.46610073180951203564437787334, 6.95992538028416343788925081002, 8.438021136232237137507525084899, 8.968018046406244119256443723761, 10.28269619456180592504330430295, 11.23040030818508390154887588662