| L(s) = 1 | + (−1.21 − 0.0635i)2-s + (3.22 + 3.97i)3-s + (−6.49 − 0.682i)4-s + (8.85 + 6.81i)5-s + (−3.65 − 5.03i)6-s + (10.7 − 15.0i)7-s + (17.4 + 2.75i)8-s + (0.160 − 0.756i)9-s + (−10.3 − 8.83i)10-s + (25.0 − 5.33i)11-s + (−18.2 − 28.0i)12-s + (2.21 + 1.12i)13-s + (−13.9 + 17.6i)14-s + (1.41 + 57.2i)15-s + (30.1 + 6.40i)16-s + (6.65 + 17.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.428 − 0.0224i)2-s + (0.620 + 0.765i)3-s + (−0.811 − 0.0852i)4-s + (0.792 + 0.609i)5-s + (−0.248 − 0.342i)6-s + (0.579 − 0.814i)7-s + (0.769 + 0.121i)8-s + (0.00595 − 0.0280i)9-s + (−0.325 − 0.279i)10-s + (0.687 − 0.146i)11-s + (−0.437 − 0.674i)12-s + (0.0471 + 0.0240i)13-s + (−0.266 + 0.336i)14-s + (0.0242 + 0.985i)15-s + (0.470 + 0.100i)16-s + (0.0950 + 0.247i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.725i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.687 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.64103 + 0.705573i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64103 + 0.705573i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-8.85 - 6.81i)T \) |
| 7 | \( 1 + (-10.7 + 15.0i)T \) |
| good | 2 | \( 1 + (1.21 + 0.0635i)T + (7.95 + 0.836i)T^{2} \) |
| 3 | \( 1 + (-3.22 - 3.97i)T + (-5.61 + 26.4i)T^{2} \) |
| 11 | \( 1 + (-25.0 + 5.33i)T + (1.21e3 - 541. i)T^{2} \) |
| 13 | \( 1 + (-2.21 - 1.12i)T + (1.29e3 + 1.77e3i)T^{2} \) |
| 17 | \( 1 + (-6.65 - 17.3i)T + (-3.65e3 + 3.28e3i)T^{2} \) |
| 19 | \( 1 + (-13.6 - 130. i)T + (-6.70e3 + 1.42e3i)T^{2} \) |
| 23 | \( 1 + (-1.98 + 37.9i)T + (-1.21e4 - 1.27e3i)T^{2} \) |
| 29 | \( 1 + (27.5 - 37.9i)T + (-7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (45.2 - 101. i)T + (-1.99e4 - 2.21e4i)T^{2} \) |
| 37 | \( 1 + (96.5 - 62.7i)T + (2.06e4 - 4.62e4i)T^{2} \) |
| 41 | \( 1 + (-463. + 150. i)T + (5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + (-91.2 - 91.2i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (-201. - 77.3i)T + (7.71e4 + 6.94e4i)T^{2} \) |
| 53 | \( 1 + (232. - 187. i)T + (3.09e4 - 1.45e5i)T^{2} \) |
| 59 | \( 1 + (313. - 348. i)T + (-2.14e4 - 2.04e5i)T^{2} \) |
| 61 | \( 1 + (-490. + 441. i)T + (2.37e4 - 2.25e5i)T^{2} \) |
| 67 | \( 1 + (142. - 54.7i)T + (2.23e5 - 2.01e5i)T^{2} \) |
| 71 | \( 1 + (557. + 405. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-108. + 167. i)T + (-1.58e5 - 3.55e5i)T^{2} \) |
| 79 | \( 1 + (528. + 1.18e3i)T + (-3.29e5 + 3.66e5i)T^{2} \) |
| 83 | \( 1 + (60.7 - 383. i)T + (-5.43e5 - 1.76e5i)T^{2} \) |
| 89 | \( 1 + (1.00e3 + 1.11e3i)T + (-7.36e4 + 7.01e5i)T^{2} \) |
| 97 | \( 1 + (-84.4 - 533. i)T + (-8.68e5 + 2.82e5i)T^{2} \) |
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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
−12.48341416802050044296175256205, −10.84943205535323643748933454558, −10.21047963491905501778713655585, −9.444148337357982271833843477075, −8.596625233756386875221916585550, −7.43216373637957795082602225309, −5.93020965300431613268351961489, −4.43383125010698695010298193138, −3.50272746605721708017582947447, −1.40166099214814650338166074264,
1.13536167196453605742406021270, 2.39607049301448930895729096845, 4.53269030032997146632199740623, 5.59088717755688551333891935591, 7.19391733006182759610561493126, 8.278411450619936149278449454068, 9.012545293731224465911568573384, 9.610450185089172640797664761405, 11.14820050028182932562657175608, 12.45097514040212413090144606350
