L(s) = 1 | + (−0.0871 − 0.996i)2-s + (1.40 + 1.01i)3-s + (−0.984 + 0.173i)4-s + (−2.23 + 0.130i)5-s + (0.887 − 1.48i)6-s + (−0.693 − 0.185i)7-s + (0.258 + 0.965i)8-s + (0.945 + 2.84i)9-s + (0.324 + 2.21i)10-s + (0.957 + 0.552i)11-s + (−1.55 − 0.754i)12-s + (1.59 + 3.42i)13-s + (−0.124 + 0.707i)14-s + (−3.26 − 2.07i)15-s + (0.939 − 0.342i)16-s + (0.308 + 3.53i)17-s + ⋯ |
L(s) = 1 | + (−0.0616 − 0.704i)2-s + (0.810 + 0.585i)3-s + (−0.492 + 0.0868i)4-s + (−0.998 + 0.0583i)5-s + (0.362 − 0.607i)6-s + (−0.262 − 0.0702i)7-s + (0.0915 + 0.341i)8-s + (0.315 + 0.948i)9-s + (0.102 + 0.699i)10-s + (0.288 + 0.166i)11-s + (−0.450 − 0.217i)12-s + (0.443 + 0.950i)13-s + (−0.0333 + 0.188i)14-s + (−0.843 − 0.536i)15-s + (0.234 − 0.0855i)16-s + (0.0749 + 0.856i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20379 + 0.564793i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20379 + 0.564793i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0871 + 0.996i)T \) |
| 3 | \( 1 + (-1.40 - 1.01i)T \) |
| 5 | \( 1 + (2.23 - 0.130i)T \) |
| 19 | \( 1 + (-0.650 - 4.31i)T \) |
good | 7 | \( 1 + (0.693 + 0.185i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.957 - 0.552i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.59 - 3.42i)T + (-8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (-0.308 - 3.53i)T + (-16.7 + 2.95i)T^{2} \) |
| 23 | \( 1 + (-0.482 - 0.337i)T + (7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (0.211 - 0.177i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.29 + 2.23i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.45 + 1.45i)T - 37iT^{2} \) |
| 41 | \( 1 + (-2.75 - 7.57i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (5.18 - 3.63i)T + (14.7 - 40.4i)T^{2} \) |
| 47 | \( 1 + (-4.28 - 0.374i)T + (46.2 + 8.16i)T^{2} \) |
| 53 | \( 1 + (3.77 + 2.64i)T + (18.1 + 49.8i)T^{2} \) |
| 59 | \( 1 + (5.58 + 4.68i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.14 + 6.46i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.128 + 1.47i)T + (-65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-14.0 - 2.47i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (2.29 + 1.06i)T + (46.9 + 55.9i)T^{2} \) |
| 79 | \( 1 + (2.66 + 7.32i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.22 - 1.66i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-5.78 - 2.10i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-15.8 + 1.38i)T + (95.5 - 16.8i)T^{2} \) |
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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
−10.89402279604625958939698983874, −9.942115712838183297410875032572, −9.220720493330660188336607157343, −8.321231295254973036819463368095, −7.69067704404888958512263021681, −6.38176098312653718040111907155, −4.76172821441940559291816773593, −3.91043692045237017027583882912, −3.29589934687525087102566542112, −1.76359114151795956438597614585,
0.74832542770664070951705119456, 2.89060825883479421122723778809, 3.79826548372638044186770229995, 5.05261977418438751074243048715, 6.36188825890858971928517462197, 7.23349197482872307592402442639, 7.82526813401090673413787161200, 8.738801857202906706767891253076, 9.275345623281283804739107637422, 10.54701364903582093675644138885