| L(s) = 1 | + (16 − 27.7i)2-s + (121.5 + 210. i)3-s + (−511. − 886. i)4-s + (−1.01e3 + 1.76e3i)5-s + 7.77e3·6-s + (4.20e4 − 1.43e4i)7-s − 3.27e4·8-s + (−2.95e4 + 5.11e4i)9-s + (3.25e4 + 5.63e4i)10-s + (3.93e4 + 6.82e4i)11-s + (1.24e5 − 2.15e5i)12-s − 7.88e5·13-s + (2.74e5 − 1.39e6i)14-s − 4.94e5·15-s + (−5.24e5 + 9.08e5i)16-s + (5.11e6 + 8.86e6i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.145 + 0.252i)5-s + 0.408·6-s + (0.946 − 0.323i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.102 + 0.178i)10-s + (0.0737 + 0.127i)11-s + (0.144 − 0.249i)12-s − 0.589·13-s + (0.136 − 0.693i)14-s − 0.168·15-s + (−0.125 + 0.216i)16-s + (0.873 + 1.51i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.87113 - 0.188250i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.87113 - 0.188250i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-16 + 27.7i)T \) |
| 3 | \( 1 + (-121.5 - 210. i)T \) |
| 7 | \( 1 + (-4.20e4 + 1.43e4i)T \) |
| good | 5 | \( 1 + (1.01e3 - 1.76e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-3.93e4 - 6.82e4i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + 7.88e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-5.11e6 - 8.86e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-1.03e7 + 1.78e7i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (3.67e6 - 6.36e6i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 - 9.93e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + (-1.00e8 - 1.74e8i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-1.95e8 + 3.38e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 - 7.80e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 2.17e7T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-1.51e9 + 2.62e9i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-1.98e9 - 3.43e9i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (1.37e9 + 2.38e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (5.25e9 - 9.10e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-1.14e9 - 1.98e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 4.54e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + (6.98e9 + 1.20e10i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (1.39e10 - 2.40e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + 6.17e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (1.20e10 - 2.09e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + 1.03e11T + 7.15e21T^{2} \) |
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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
−13.69858312134191462454234227369, −12.25233544733534854520067027773, −11.09372651551942209329918306821, −10.19502153141529029034360288206, −8.802598330510709171431069360741, −7.35708947958100768216398341170, −5.32963680894483250784248038604, −4.17157637799142891905223780014, −2.77235282657427506259652257314, −1.16069158055762352552259325015,
0.969477061125037627392974048892, 2.76213845762736753039474071513, 4.58428872110303287702798195760, 5.83555044013265365341735711571, 7.49250789051589596135050414750, 8.222950499768543344888403220359, 9.702525059474743708850485176825, 11.71611436672429472415618496579, 12.39559986060462363481336347716, 14.02533233074674697531412087260
