| L(s) = 1 | + (−2.96 + 9.11i)3-s + (−76.5 + 55.6i)5-s + (16.4 + 50.7i)7-s + (122. + 88.8i)9-s + (−75.5 − 394. i)11-s + (−595. − 432. i)13-s + (−280. − 862. i)15-s + (842. − 611. i)17-s + (−171. + 528. i)19-s − 511.·21-s − 812.·23-s + (1.80e3 − 5.54e3i)25-s + (−3.05e3 + 2.22e3i)27-s + (−1.41e3 − 4.35e3i)29-s + (−2.12e3 − 1.54e3i)31-s + ⋯ |
| L(s) = 1 | + (−0.189 + 0.584i)3-s + (−1.36 + 0.995i)5-s + (0.127 + 0.391i)7-s + (0.503 + 0.365i)9-s + (−0.188 − 0.982i)11-s + (−0.976 − 0.709i)13-s + (−0.321 − 0.989i)15-s + (0.706 − 0.513i)17-s + (−0.109 + 0.335i)19-s − 0.252·21-s − 0.320·23-s + (0.576 − 1.77i)25-s + (−0.806 + 0.586i)27-s + (−0.312 − 0.960i)29-s + (−0.396 − 0.287i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.403 + 0.915i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.403 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0449251 - 0.0688954i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0449251 - 0.0688954i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + (75.5 + 394. i)T \) |
| good | 3 | \( 1 + (2.96 - 9.11i)T + (-196. - 142. i)T^{2} \) |
| 5 | \( 1 + (76.5 - 55.6i)T + (965. - 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-16.4 - 50.7i)T + (-1.35e4 + 9.87e3i)T^{2} \) |
| 13 | \( 1 + (595. + 432. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-842. + 611. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (171. - 528. i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + 812.T + 6.43e6T^{2} \) |
| 29 | \( 1 + (1.41e3 + 4.35e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (2.12e3 + 1.54e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (4.49e3 + 1.38e4i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (4.57e3 - 1.40e4i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 8.82e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (7.84e3 - 2.41e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (3.32e3 + 2.41e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (3.50e3 + 1.07e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.33e4 + 9.70e3i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + 3.14e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (6.32e4 - 4.59e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (1.78e3 + 5.48e3i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-4.16e4 - 3.02e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (7.56e4 - 5.49e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + 5.43e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (6.42e4 + 4.67e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−12.64901849142143732371848737828, −11.55236867032450139880704865056, −10.80884605507376929756911559034, −9.788033646291174891264551544371, −8.061194388084623929409124197627, −7.34671908413772808646914057296, −5.60240947363577250494900829680, −4.12539363055574200714514867535, −2.89522763418898243401528687745, −0.03592209939682580335298576954,
1.47069078573274424359468428103, 3.94097643538536485314528134862, 4.94504315643526154313705458414, 7.02440722532360489336732201761, 7.63999042341091572831194396021, 8.946076770912336887068419008065, 10.27955884719984287134371299165, 11.98051680201982029666148415112, 12.18109048146609217147940515220, 13.19874944223719285137496241161
