| L(s) = 1 | + (0.0678 + 3.99i)2-s + (−2.72 − 8.57i)3-s + (−15.9 + 0.542i)4-s + (16.6 − 28.7i)5-s + (34.1 − 11.4i)6-s + (39.9 − 23.0i)7-s + (−3.25 − 63.9i)8-s + (−66.0 + 46.8i)9-s + (116. + 64.4i)10-s + (−63.6 + 36.7i)11-s + (48.3 + 135. i)12-s + (151. − 262. i)13-s + (95.0 + 158. i)14-s + (−292. − 63.9i)15-s + (255. − 17.3i)16-s − 182.·17-s + ⋯ |
| L(s) = 1 | + (0.0169 + 0.999i)2-s + (−0.303 − 0.952i)3-s + (−0.999 + 0.0339i)4-s + (0.664 − 1.15i)5-s + (0.947 − 0.319i)6-s + (0.815 − 0.471i)7-s + (−0.0508 − 0.998i)8-s + (−0.816 + 0.578i)9-s + (1.16 + 0.644i)10-s + (−0.525 + 0.303i)11-s + (0.335 + 0.942i)12-s + (0.896 − 1.55i)13-s + (0.484 + 0.807i)14-s + (−1.29 − 0.284i)15-s + (0.997 − 0.0677i)16-s − 0.629·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.19293 - 0.466837i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19293 - 0.466837i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0678 - 3.99i)T \) |
| 3 | \( 1 + (2.72 + 8.57i)T \) |
| good | 5 | \( 1 + (-16.6 + 28.7i)T + (-312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + (-39.9 + 23.0i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (63.6 - 36.7i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-151. + 262. i)T + (-1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 + 182.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 314. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (290. + 167. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-357. - 618. i)T + (-3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-985. - 568. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 1.00e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (557. - 965. i)T + (-1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-2.18e3 + 1.25e3i)T + (1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-980. + 566. i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + 1.05e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (-878. - 507. i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (430. + 745. i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-559. - 322. i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + 9.56e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 1.89e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (6.76e3 - 3.90e3i)T + (1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (7.05e3 - 4.07e3i)T + (2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 - 7.65e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (6.36e3 + 1.10e4i)T + (-4.42e7 + 7.66e7i)T^{2} \) |
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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
−15.75822099686395140292180257059, −14.16643096125612461322267057609, −13.24334572411813404676824182801, −12.50184295970693700246812225309, −10.46688446261001454435505342093, −8.568093630748380856491417564852, −7.80321504889808466476214772532, −6.03307653315911232720596598675, −4.95088404849257255972515980759, −1.03695869917525662800809481985,
2.49368967997991338476370962875, 4.37709642359552229900872195901, 6.05090264148226486824323572110, 8.698392110534094773159176398486, 9.871982508350785666652952500776, 11.08479776103311377472132780014, 11.52625029750906959067172169713, 13.65397166174841104275989783851, 14.42448992356364265889688938423, 15.68440259353847352997467244417
