| L(s) = 1 | + (−16 − 27.7i)2-s + (−94.7 + 164. i)3-s + (−511. + 886. i)4-s + (−3.21e3 − 5.56e3i)5-s + 6.06e3·6-s + (−9.57e3 + 4.34e4i)7-s + 3.27e4·8-s + (7.06e4 + 1.22e5i)9-s + (−1.02e5 + 1.77e5i)10-s + (4.59e5 − 7.96e5i)11-s + (−9.70e4 − 1.68e5i)12-s + 1.22e6·13-s + (1.35e6 − 4.29e5i)14-s + 1.21e6·15-s + (−5.24e5 − 9.08e5i)16-s + (3.31e6 − 5.73e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.225 + 0.389i)3-s + (−0.249 + 0.433i)4-s + (−0.459 − 0.795i)5-s + 0.318·6-s + (−0.215 + 0.976i)7-s + 0.353·8-s + (0.398 + 0.690i)9-s + (−0.324 + 0.562i)10-s + (0.861 − 1.49i)11-s + (−0.112 − 0.194i)12-s + 0.917·13-s + (0.674 − 0.213i)14-s + 0.413·15-s + (−0.125 − 0.216i)16-s + (0.565 − 0.979i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.276i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.960 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.28997 - 0.182058i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28997 - 0.182058i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (16 + 27.7i)T \) |
| 7 | \( 1 + (9.57e3 - 4.34e4i)T \) |
| good | 3 | \( 1 + (94.7 - 164. i)T + (-8.85e4 - 1.53e5i)T^{2} \) |
| 5 | \( 1 + (3.21e3 + 5.56e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-4.59e5 + 7.96e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 - 1.22e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-3.31e6 + 5.73e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-9.33e6 - 1.61e7i)T + (-5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-1.43e7 - 2.48e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + 7.73e6T + 1.22e16T^{2} \) |
| 31 | \( 1 + (5.68e7 - 9.85e7i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-9.50e7 - 1.64e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 + 1.74e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.83e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (5.75e8 + 9.97e8i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (3.18e8 - 5.51e8i)T + (-4.63e18 - 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-1.39e9 + 2.41e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-2.60e9 - 4.50e9i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-3.19e9 + 5.52e9i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 - 2.41e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (-8.18e9 + 1.41e10i)T + (-1.56e20 - 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-1.91e9 - 3.31e9i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + 2.84e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-1.35e10 - 2.34e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + 1.49e11T + 7.15e21T^{2} \) |
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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
−16.48689388031605398942268311684, −16.05257573492397625563693052526, −13.81217900285311039452448791512, −12.23046770432484284333225615148, −11.19404538596459213210795145481, −9.406367302845865680628875206902, −8.221808558578387884621114596433, −5.48889291722345986632416675528, −3.54226460854227878467960621946, −1.10316622732432792463937293850,
0.993314441776735800178329016825, 3.98502386243268892342293191685, 6.61596829530497141909373675048, 7.35618475081384554659872749820, 9.531596603666221686064297599624, 11.05058314140766736082926861260, 12.80997882574838300491251345149, 14.48586945480261266268219794288, 15.52245991919417532021251794864, 17.15113561992471658251111103095
