| L(s) = 1 | + (−2.46 − 0.661i)2-s + (−7.06 − 26.3i)3-s + (−22.0 − 12.7i)4-s + (47.7 − 29.0i)5-s + 69.7i·6-s + (−83.3 + 99.2i)7-s + (103. + 103. i)8-s + (−435. + 251. i)9-s + (−137. + 40.2i)10-s + (344. − 596. i)11-s + (−180. + 671. i)12-s + (−368. + 368. i)13-s + (271. − 189. i)14-s + (−1.10e3 − 1.05e3i)15-s + (219. + 380. i)16-s + (−671. + 179. i)17-s + ⋯ |
| L(s) = 1 | + (−0.436 − 0.116i)2-s + (−0.453 − 1.69i)3-s + (−0.689 − 0.397i)4-s + (0.854 − 0.520i)5-s + 0.791i·6-s + (−0.643 + 0.765i)7-s + (0.573 + 0.573i)8-s + (−1.79 + 1.03i)9-s + (−0.433 + 0.127i)10-s + (0.858 − 1.48i)11-s + (−0.360 + 1.34i)12-s + (−0.604 + 0.604i)13-s + (0.370 − 0.258i)14-s + (−1.26 − 1.20i)15-s + (0.214 + 0.371i)16-s + (−0.563 + 0.150i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.162117 + 0.544779i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.162117 + 0.544779i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-47.7 + 29.0i)T \) |
| 7 | \( 1 + (83.3 - 99.2i)T \) |
| good | 2 | \( 1 + (2.46 + 0.661i)T + (27.7 + 16i)T^{2} \) |
| 3 | \( 1 + (7.06 + 26.3i)T + (-210. + 121.5i)T^{2} \) |
| 11 | \( 1 + (-344. + 596. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (368. - 368. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (671. - 179. i)T + (1.22e6 - 7.09e5i)T^{2} \) |
| 19 | \( 1 + (152. + 264. i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-60.6 + 226. i)T + (-5.57e6 - 3.21e6i)T^{2} \) |
| 29 | \( 1 + 1.30e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + (6.32e3 + 3.64e3i)T + (1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (6.68e3 + 1.79e3i)T + (6.00e7 + 3.46e7i)T^{2} \) |
| 41 | \( 1 - 2.92e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (960. + 960. i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (-188. + 702. i)T + (-1.98e8 - 1.14e8i)T^{2} \) |
| 53 | \( 1 + (1.59e4 - 4.28e3i)T + (3.62e8 - 2.09e8i)T^{2} \) |
| 59 | \( 1 + (-7.78e3 + 1.34e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-1.03e4 + 5.98e3i)T + (4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-182. - 682. i)T + (-1.16e9 + 6.75e8i)T^{2} \) |
| 71 | \( 1 - 6.20e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-6.91e3 - 2.58e4i)T + (-1.79e9 + 1.03e9i)T^{2} \) |
| 79 | \( 1 + (-7.29e4 + 4.21e4i)T + (1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (8.57e3 - 8.57e3i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + (5.21e4 + 9.03e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (3.02e4 + 3.02e4i)T + 8.58e9iT^{2} \) |
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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
−14.25200953170444741420802461740, −13.46171873499018028080481806992, −12.56363541063881492236794317268, −11.27652170922794266436976861357, −9.348499338181261115269763618156, −8.486494686861703087051220417082, −6.49394708912998258450930823944, −5.56667217609278859311872251361, −1.90674801932169326764609995072, −0.41309997897277440781189114740,
3.68687287528318459858309788900, 4.96840076051770561210939621421, 6.94664546878494941964545814369, 9.262465886194016291323101310648, 9.837342183135542228658341740308, 10.65007996803218489881009440598, 12.58743648348543805606373215228, 14.11790691130197784985785434742, 15.15899334871410304056902308127, 16.51071989187679088050140085375
