| L(s) = 1 | + (−0.740 − 2.28i)2-s + (−21.7 − 15.8i)3-s + (21.2 − 15.4i)4-s + (−5.23 + 16.1i)5-s + (−19.9 + 61.3i)6-s + (87.7 − 63.7i)7-s + (−112. − 82.0i)8-s + (148. + 456. i)9-s + 40.6·10-s + (391. − 89.5i)11-s − 705.·12-s + (−235. − 723. i)13-s + (−210. − 152. i)14-s + (368. − 267. i)15-s + (156. − 480. i)16-s + (−490. + 1.51e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.130 − 0.403i)2-s + (−1.39 − 1.01i)3-s + (0.663 − 0.482i)4-s + (−0.0937 + 0.288i)5-s + (−0.225 + 0.695i)6-s + (0.676 − 0.491i)7-s + (−0.624 − 0.453i)8-s + (0.610 + 1.87i)9-s + 0.128·10-s + (0.974 − 0.223i)11-s − 1.41·12-s + (−0.385 − 1.18i)13-s + (−0.286 − 0.208i)14-s + (0.423 − 0.307i)15-s + (0.152 − 0.469i)16-s + (−0.412 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.466891 - 0.743204i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.466891 - 0.743204i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-391. + 89.5i)T \) |
| good | 2 | \( 1 + (0.740 + 2.28i)T + (-25.8 + 18.8i)T^{2} \) |
| 3 | \( 1 + (21.7 + 15.8i)T + (75.0 + 231. i)T^{2} \) |
| 5 | \( 1 + (5.23 - 16.1i)T + (-2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-87.7 + 63.7i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (235. + 723. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (490. - 1.51e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-950. - 690. i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 - 1.15e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (1.97e3 - 1.43e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (577. + 1.77e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-1.02e4 + 7.45e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (2.41e3 + 1.75e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 3.67e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (680. + 494. i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-5.51e3 - 1.69e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-8.58e3 + 6.23e3i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (1.03e4 - 3.18e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 - 4.49e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + (3.71e3 - 1.14e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (6.47e4 - 4.70e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-8.05e3 - 2.47e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-2.12e4 + 6.55e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 - 1.56e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-2.23e4 - 6.87e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−18.99918138426975538815562681861, −17.78726712955949242126032252097, −16.74822199621387711191234688043, −14.81291277203510866879900048953, −12.73407516307314478265256340205, −11.44617231776132500818400400946, −10.61600754674900464059391539730, −7.29788807395273456564988677703, −5.85297423824834172950025332653, −1.19497711822309197012057287721,
4.81128720332061435139428446125, 6.69731895382502131916332336372, 9.245565864143789459988912472913, 11.42888598970056716623957416436, 11.86713370117616697849567660749, 14.88542303124410871605016657824, 16.16647820396133025944080806486, 16.88012097493825478068833523026, 17.99760413930491118581317265031, 20.39469505425446940580706696687
