| L(s) = 1 | + (−25.8 − 18.8i)2-s + (−176. + 543. i)3-s + (316. + 973. i)4-s + (3.17e3 − 2.30e3i)5-s + (1.48e4 − 1.07e4i)6-s + (−1.83e4 − 5.64e4i)7-s + (1.01e4 − 3.11e4i)8-s + (−1.21e5 − 8.80e4i)9-s − 1.25e5·10-s + (3.93e5 + 3.61e5i)11-s − 5.85e5·12-s + (−2.63e5 − 1.91e5i)13-s + (−5.86e5 + 1.80e6i)14-s + (6.93e5 + 2.13e6i)15-s + (−8.48e5 + 6.16e5i)16-s + (4.44e6 − 3.22e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.419 + 1.29i)3-s + (0.154 + 0.475i)4-s + (0.454 − 0.330i)5-s + (0.777 − 0.564i)6-s + (−0.412 − 1.26i)7-s + (0.109 − 0.336i)8-s + (−0.684 − 0.497i)9-s − 0.397·10-s + (0.736 + 0.676i)11-s − 0.679·12-s + (−0.196 − 0.142i)13-s + (−0.291 + 0.897i)14-s + (0.235 + 0.725i)15-s + (−0.202 + 0.146i)16-s + (0.758 − 0.551i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.22526 + 0.337794i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22526 + 0.337794i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (25.8 + 18.8i)T \) |
| 11 | \( 1 + (-3.93e5 - 3.61e5i)T \) |
| good | 3 | \( 1 + (176. - 543. i)T + (-1.43e5 - 1.04e5i)T^{2} \) |
| 5 | \( 1 + (-3.17e3 + 2.30e3i)T + (1.50e7 - 4.64e7i)T^{2} \) |
| 7 | \( 1 + (1.83e4 + 5.64e4i)T + (-1.59e9 + 1.16e9i)T^{2} \) |
| 13 | \( 1 + (2.63e5 + 1.91e5i)T + (5.53e11 + 1.70e12i)T^{2} \) |
| 17 | \( 1 + (-4.44e6 + 3.22e6i)T + (1.05e13 - 3.25e13i)T^{2} \) |
| 19 | \( 1 + (3.78e5 - 1.16e6i)T + (-9.42e13 - 6.84e13i)T^{2} \) |
| 23 | \( 1 - 3.88e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (-5.53e7 - 1.70e8i)T + (-9.87e15 + 7.17e15i)T^{2} \) |
| 31 | \( 1 + (-1.14e8 - 8.32e7i)T + (7.85e15 + 2.41e16i)T^{2} \) |
| 37 | \( 1 + (3.17e7 + 9.78e7i)T + (-1.43e17 + 1.04e17i)T^{2} \) |
| 41 | \( 1 + (-1.73e8 + 5.32e8i)T + (-4.45e17 - 3.23e17i)T^{2} \) |
| 43 | \( 1 - 1.21e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (3.44e8 - 1.06e9i)T + (-2.00e18 - 1.45e18i)T^{2} \) |
| 53 | \( 1 + (3.42e9 + 2.48e9i)T + (2.86e18 + 8.81e18i)T^{2} \) |
| 59 | \( 1 + (-1.17e9 - 3.60e9i)T + (-2.43e19 + 1.77e19i)T^{2} \) |
| 61 | \( 1 + (-5.16e9 + 3.75e9i)T + (1.34e19 - 4.13e19i)T^{2} \) |
| 67 | \( 1 - 1.97e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + (-6.22e9 + 4.52e9i)T + (7.14e19 - 2.19e20i)T^{2} \) |
| 73 | \( 1 + (-4.18e9 - 1.28e10i)T + (-2.53e20 + 1.84e20i)T^{2} \) |
| 79 | \( 1 + (-3.96e8 - 2.88e8i)T + (2.31e20 + 7.11e20i)T^{2} \) |
| 83 | \( 1 + (-5.62e10 + 4.08e10i)T + (3.97e20 - 1.22e21i)T^{2} \) |
| 89 | \( 1 + 7.89e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (8.23e10 + 5.98e10i)T + (2.21e21 + 6.80e21i)T^{2} \) |
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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
−15.91916867962747151864866717769, −14.30085392952677457471520126010, −12.67228458389419612313575729730, −11.03183115669732028859734880752, −10.04746086614501722980443423453, −9.269319274598177569406926333260, −7.10193586647969767652970627605, −4.92347417805234321732776581089, −3.54859596774208965386576511792, −1.01805929915380217423150875555,
0.859911876010302938119400786909, 2.40716467280994405802285940731, 5.87652263899145739200760778291, 6.55574839374034517032599235739, 8.178915400672277029035679971805, 9.598845087358189348378956183711, 11.48627398276816639652019720389, 12.56344658279129636519600977472, 13.96277474551562790914604373751, 15.34497520674666633708210425836
