| L(s) = 1 | + (−3.23 + 2.35i)2-s + (2.78 − 8.55i)3-s + (4.94 − 15.2i)4-s + (49.1 − 26.5i)5-s + (11.1 + 34.2i)6-s − 149.·7-s + (19.7 + 60.8i)8-s + (−65.5 − 47.6i)9-s + (−96.7 + 201. i)10-s + (502. − 364. i)11-s + (−116. − 84.6i)12-s + (−570. − 414. i)13-s + (483. − 351. i)14-s + (−90.5 − 494. i)15-s + (−207. − 150. i)16-s + (−211. − 651. i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (0.879 − 0.475i)5-s + (0.126 + 0.388i)6-s − 1.15·7-s + (0.109 + 0.336i)8-s + (−0.269 − 0.195i)9-s + (−0.305 + 0.637i)10-s + (1.25 − 0.908i)11-s + (−0.233 − 0.169i)12-s + (−0.935 − 0.679i)13-s + (0.659 − 0.479i)14-s + (−0.103 − 0.567i)15-s + (−0.202 − 0.146i)16-s + (−0.177 − 0.547i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7748683758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7748683758\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.23 - 2.35i)T \) |
| 3 | \( 1 + (-2.78 + 8.55i)T \) |
| 5 | \( 1 + (-49.1 + 26.5i)T \) |
| good | 7 | \( 1 + 149.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-502. + 364. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (570. + 414. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (211. + 651. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-651. - 2.00e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (1.78e3 - 1.29e3i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (1.68e3 - 5.18e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (1.39e3 + 4.28e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (4.05e3 + 2.94e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (1.20e4 + 8.71e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 7.81e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-7.13e3 + 2.19e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.19e4 - 3.66e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (3.49e4 + 2.53e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-1.40e4 + 1.01e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (7.86e3 + 2.42e4i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (8.48e3 - 2.61e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (3.58e4 - 2.60e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (1.48e3 - 4.56e3i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (3.31e4 + 1.02e5i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (5.96e4 - 4.33e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-6.25e3 + 1.92e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−11.85745236983124516450131085779, −10.24761864928791591708637932029, −9.472741204997859300516098063957, −8.699021961721181397374074276171, −7.36301131692127607439359096612, −6.30030914832293899064656186636, −5.52173091354600045857291333150, −3.34459736568004576575340056291, −1.68025217963578700288258249553, −0.29193396834044731324438428238,
1.88912345347939007486048114097, 3.10762542461791261150538622477, 4.51462143720750989900623464674, 6.35902968521626589732033322038, 7.08213960620502670766023968558, 8.910981034321825947006374958459, 9.702447033697343242867526072632, 10.03337765865296123985760161904, 11.39258794493916633269473813501, 12.40043359479434835329974743083
