| L(s) = 1 | + (−0.939 + 0.342i)2-s + (1.22 − 1.22i)3-s + (0.766 − 0.642i)4-s + (−0.0262 − 0.148i)5-s + (−0.736 + 1.56i)6-s + (−0.766 − 0.642i)7-s + (−0.500 + 0.866i)8-s + (0.0176 − 2.99i)9-s + (0.0755 + 0.130i)10-s + (0.400 − 2.27i)11-s + (0.156 − 1.72i)12-s + (3.71 + 1.35i)13-s + (0.939 + 0.342i)14-s + (−0.213 − 0.150i)15-s + (0.173 − 0.984i)16-s + (−2.15 − 3.73i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (0.709 − 0.705i)3-s + (0.383 − 0.321i)4-s + (−0.0117 − 0.0665i)5-s + (−0.300 + 0.639i)6-s + (−0.289 − 0.242i)7-s + (−0.176 + 0.306i)8-s + (0.00586 − 0.999i)9-s + (0.0238 + 0.0413i)10-s + (0.120 − 0.684i)11-s + (0.0450 − 0.497i)12-s + (1.03 + 0.375i)13-s + (0.251 + 0.0914i)14-s + (−0.0552 − 0.0389i)15-s + (0.0434 − 0.246i)16-s + (−0.522 − 0.905i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.991616 - 0.679452i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.991616 - 0.679452i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
| 7 | \( 1 + (0.766 + 0.642i)T \) |
| good | 5 | \( 1 + (0.0262 + 0.148i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-0.400 + 2.27i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-3.71 - 1.35i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.15 + 3.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.94 + 3.37i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.37 - 4.50i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.393 - 0.143i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.824 + 0.692i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (1.42 + 2.47i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.55 + 0.565i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.156 + 0.885i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.47 - 2.91i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + (-1.29 - 7.34i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-8.81 - 7.39i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-5.44 - 1.98i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (2.74 + 4.75i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.41 - 5.91i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (13.0 - 4.75i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (12.5 - 4.57i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-1.95 + 3.38i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.21 - 12.5i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.25319982224869422200491196196, −10.05055781794051364438020736101, −9.014416424904675843953910354083, −8.591882076232718785211969321472, −7.42604154307823395169532700596, −6.73895725151944372552906515913, −5.71798279460458549893239162915, −3.88681675469611882046288446366, −2.60227951228864047575455702461, −0.989549480616510056379259740138,
1.93136760735043808730805178594, 3.28355238290177368491226948580, 4.27716647623819413668117135126, 5.80463018334417399145080174730, 7.01920925040704332571029851082, 8.304417451989979406093840608306, 8.654010602050425558198708671086, 9.850742108500759667695176356682, 10.33315307773947694497896327072, 11.23524181679702600247445713597
