L(s) = 1 | + (−5.65 − 0.181i)2-s + (31.9 + 2.04i)4-s + 69.1i·5-s + 238. i·7-s + (−180. − 17.3i)8-s + (12.5 − 391. i)10-s − 350.·11-s + 669.·13-s + (43.2 − 1.35e3i)14-s + (1.01e3 + 130. i)16-s + 1.39e3i·17-s − 1.78e3i·19-s + (−141. + 2.20e3i)20-s + (1.98e3 + 63.5i)22-s − 1.32e3·23-s + ⋯ |
L(s) = 1 | + (−0.999 − 0.0320i)2-s + (0.997 + 0.0640i)4-s + 1.23i·5-s + 1.84i·7-s + (−0.995 − 0.0959i)8-s + (0.0396 − 1.23i)10-s − 0.874·11-s + 1.09·13-s + (0.0589 − 1.84i)14-s + (0.991 + 0.127i)16-s + 1.17i·17-s − 1.13i·19-s + (−0.0792 + 1.23i)20-s + (0.874 + 0.0280i)22-s − 0.521·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0640i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.0230169 + 0.718514i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0230169 + 0.718514i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (5.65 + 0.181i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 69.1iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 238. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 350.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 669.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.39e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.78e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 1.32e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.89e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 2.33e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 1.29e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.71e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.06e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 3.98e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.05e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.51e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.96e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.24e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 6.50e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.58e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.74e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 4.86e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.93e3iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 5.65e4T + 8.58e9T^{2} \) |
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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
−13.08896670256640312863443142704, −11.84407087428610364479036636587, −11.03997184380698741123743577871, −10.12908096096134825665155155488, −8.854689352010842991501450313557, −8.060718194596370991810999215758, −6.56776392603027890980293987645, −5.76741674102834234786256319458, −3.07707879279604105124227499357, −2.11534485428649399393689800776,
0.38632454356948183721764351549, 1.40239142103466940700385007682, 3.71069326688308417993925828118, 5.32748157944248904147846424371, 6.98998066886707370416876439210, 7.942793212232242598979908874586, 8.897895874995991701722827756470, 10.16430609285008301630644164663, 10.81331794570301401392656606296, 12.14781188372673535969264176690