| L(s) = 1 | + (1.30 − 4.00i)2-s + (−7.28 + 5.29i)3-s + (11.5 + 8.37i)4-s + (−26.9 − 83.0i)5-s + (11.7 + 36.0i)6-s + (−114. − 82.8i)7-s + (157. − 114. i)8-s + (25.0 − 77.0i)9-s − 367.·10-s + (−279. − 288. i)11-s − 128.·12-s + (−100. + 307. i)13-s + (−480. + 348. i)14-s + (635. + 461. i)15-s + (−112. − 346. i)16-s + (59.5 + 183. i)17-s + ⋯ |
| L(s) = 1 | + (0.230 − 0.708i)2-s + (−0.467 + 0.339i)3-s + (0.360 + 0.261i)4-s + (−0.482 − 1.48i)5-s + (0.132 + 0.408i)6-s + (−0.879 − 0.638i)7-s + (0.870 − 0.632i)8-s + (0.103 − 0.317i)9-s − 1.16·10-s + (−0.696 − 0.717i)11-s − 0.257·12-s + (−0.164 + 0.505i)13-s + (−0.654 + 0.475i)14-s + (0.729 + 0.529i)15-s + (−0.110 − 0.338i)16-s + (0.0500 + 0.153i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 + 0.728i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.490836 - 1.13602i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.490836 - 1.13602i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (7.28 - 5.29i)T \) |
| 11 | \( 1 + (279. + 288. i)T \) |
| good | 2 | \( 1 + (-1.30 + 4.00i)T + (-25.8 - 18.8i)T^{2} \) |
| 5 | \( 1 + (26.9 + 83.0i)T + (-2.52e3 + 1.83e3i)T^{2} \) |
| 7 | \( 1 + (114. + 82.8i)T + (5.19e3 + 1.59e4i)T^{2} \) |
| 13 | \( 1 + (100. - 307. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-59.5 - 183. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-1.35e3 + 985. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 - 3.66e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-1.17e3 - 852. i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (1.29e3 - 3.98e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (1.07e4 + 7.82e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (9.38e3 - 6.81e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.33e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.49e4 + 1.08e4i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-7.73e3 + 2.38e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (1.89e4 + 1.37e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-1.51e4 - 4.67e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 - 6.09e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (5.65e3 + 1.74e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (3.43e4 + 2.49e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-7.67e3 + 2.36e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (3.22e4 + 9.91e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 - 2.98e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (3.62e3 - 1.11e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−15.86707064414630338181571954660, −13.43632664414761138652066565193, −12.67050019902181394650969804556, −11.62824486195161438815859667694, −10.44016023128082502305048868376, −8.932541649838507604381513558249, −7.15799381481894636774215025634, −5.00386662369594108564225479546, −3.51384287819479999175385670771, −0.74109427050550311412991097151,
2.77029607154986331552728582296, 5.52018806408320623539453073506, 6.77759693733697733932927975309, 7.57072072076464249819034296520, 10.09632219446077890421264186112, 11.11738902852453541378918854307, 12.44615662194641002295415396430, 14.01291813049646500386675979888, 15.33691441746219385256728748624, 15.62966793768669763136853301265
