L(s) = 1 | + (0.919 − 1.07i)2-s + (−0.309 − 1.97i)4-s + (−0.509 + 2.56i)5-s + (−1.78 + 4.31i)7-s + (−2.40 − 1.48i)8-s + (2.28 + 2.90i)10-s + (0.337 − 0.225i)11-s + (0.558 + 2.80i)13-s + (2.99 + 5.88i)14-s + (−3.80 + 1.22i)16-s + (−2.50 + 2.50i)17-s + (−2.54 + 0.506i)19-s + (5.22 + 0.213i)20-s + (0.0678 − 0.569i)22-s + (2.78 − 1.15i)23-s + ⋯ |
L(s) = 1 | + (0.650 − 0.759i)2-s + (−0.154 − 0.987i)4-s + (−0.228 + 1.14i)5-s + (−0.675 + 1.63i)7-s + (−0.851 − 0.524i)8-s + (0.722 + 0.918i)10-s + (0.101 − 0.0679i)11-s + (0.154 + 0.778i)13-s + (0.799 + 1.57i)14-s + (−0.951 + 0.306i)16-s + (−0.607 + 0.607i)17-s + (−0.584 + 0.116i)19-s + (1.16 + 0.0476i)20-s + (0.0144 − 0.121i)22-s + (0.580 − 0.240i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.27251 + 0.660494i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.27251 + 0.660494i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.919 + 1.07i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.509 - 2.56i)T + (-4.61 - 1.91i)T^{2} \) |
| 7 | \( 1 + (1.78 - 4.31i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.337 + 0.225i)T + (4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-0.558 - 2.80i)T + (-12.0 + 4.97i)T^{2} \) |
| 17 | \( 1 + (2.50 - 2.50i)T - 17iT^{2} \) |
| 19 | \( 1 + (2.54 - 0.506i)T + (17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-2.78 + 1.15i)T + (16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (-4.40 - 2.94i)T + (11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + 0.289iT - 31T^{2} \) |
| 37 | \( 1 + (-1.93 - 0.384i)T + (34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-5.97 + 2.47i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (3.47 + 5.19i)T + (-16.4 + 39.7i)T^{2} \) |
| 47 | \( 1 + (0.140 - 0.140i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.438 - 0.292i)T + (20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-1.05 + 5.31i)T + (-54.5 - 22.5i)T^{2} \) |
| 61 | \( 1 + (-4.77 + 7.14i)T + (-23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 + (-2.53 + 3.79i)T + (-25.6 - 61.8i)T^{2} \) |
| 71 | \( 1 + (5.35 - 12.9i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-5.89 - 14.2i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (4.17 + 4.17i)T + 79iT^{2} \) |
| 83 | \( 1 + (7.83 - 1.55i)T + (76.6 - 31.7i)T^{2} \) |
| 89 | \( 1 + (-6.46 - 2.67i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 - 16.2iT - 97T^{2} \) |
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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
−11.04059784157824493342110002387, −10.19938639186472662032484009082, −9.220725280433813401036661896885, −8.554564034348864133875787690419, −6.73446938380577870109704495221, −6.36418067706672476741479288043, −5.29160083764666896213394549125, −3.96715380603846047982181459070, −2.92311106222334297438484396585, −2.15478901697543003945486408019,
0.64910947268555341434197516545, 3.10155763272937264344802504751, 4.26334178318478338193067240974, 4.75970768810336153596363197335, 6.06399758897815333319158800627, 6.98497958447426302936666904149, 7.76216266578586012600251841441, 8.629316102157809383228091805883, 9.546445028811446465312650252184, 10.61619131610242477808762443540