| L(s) = 1 | + (1.30 + 0.751i)2-s + (−14.8 − 25.7i)4-s + (8.15 − 14.1i)5-s + (−67.6 + 110. i)7-s − 92.8i·8-s + (21.2 − 12.2i)10-s + (−558. + 322. i)11-s + 524. i·13-s + (−171. + 93.1i)14-s + (−405. + 703. i)16-s + (−1.00e3 − 1.73e3i)17-s + (−421. − 243. i)19-s − 484.·20-s − 969.·22-s + (−1.45e3 − 840. i)23-s + ⋯ |
| L(s) = 1 | + (0.230 + 0.132i)2-s + (−0.464 − 0.804i)4-s + (0.145 − 0.252i)5-s + (−0.521 + 0.853i)7-s − 0.512i·8-s + (0.0671 − 0.0387i)10-s + (−1.39 + 0.803i)11-s + 0.861i·13-s + (−0.233 + 0.127i)14-s + (−0.396 + 0.686i)16-s + (−0.842 − 1.46i)17-s + (−0.268 − 0.154i)19-s − 0.271·20-s − 0.427·22-s + (−0.573 − 0.331i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.00797878 + 0.0650658i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00797878 + 0.0650658i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (67.6 - 110. i)T \) |
| good | 2 | \( 1 + (-1.30 - 0.751i)T + (16 + 27.7i)T^{2} \) |
| 5 | \( 1 + (-8.15 + 14.1i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (558. - 322. i)T + (8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 - 524. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + (1.00e3 + 1.73e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (421. + 243. i)T + (1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (1.45e3 + 840. i)T + (3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + 1.65e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + (6.39e3 - 3.69e3i)T + (1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-4.48e3 + 7.76e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 - 1.43e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.61e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-2.30e3 + 4.00e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (332. - 191. i)T + (2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (6.03e3 + 1.04e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-1.00e4 - 5.81e3i)T + (4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (7.39e3 + 1.28e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 5.60e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (5.36e4 - 3.09e4i)T + (1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (7.04e3 - 1.22e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 1.20e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + (4.22e4 - 7.32e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.28e5iT - 8.58e9T^{2} \) |
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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
−14.49204846118039028988470106450, −13.33804202574628420259127254575, −12.54904239050281112058554503418, −11.01580420824244148739199447303, −9.672325377401031232469945984950, −8.974680349423064770558533921330, −7.08295595342730714478401045397, −5.64654183665067161675593719819, −4.62729950057238994831433519997, −2.29215200650291444569421290666,
0.02668992831001728962199826026, 2.86225012986285890752317934693, 4.15148232555620727822091324772, 5.89357975670478176630875316719, 7.59249254077667631236810906598, 8.519716597882561519318460691911, 10.20887008444003834451204522570, 11.05024383793816117065987540363, 12.90160186547762274684895100202, 13.07288782556261030255255416162
