| 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
−12.45097514040212413090144606350, −11.14820050028182932562657175608, −9.610450185089172640797664761405, −9.012545293731224465911568573384, −8.278411450619936149278449454068, −7.19391733006182759610561493126, −5.59088717755688551333891935591, −4.53269030032997146632199740623, −2.39607049301448930895729096845, −1.13536167196453605742406021270,
1.40166099214814650338166074264, 3.50272746605721708017582947447, 4.43383125010698695010298193138, 5.93020965300431613268351961489, 7.43216373637957795082602225309, 8.596625233756386875221916585550, 9.444148337357982271833843477075, 10.21047963491905501778713655585, 10.84943205535323643748933454558, 12.48341416802050044296175256205
