L(s) = 1 | + (−0.415 + 0.909i)2-s + (−0.902 + 0.579i)3-s + (−0.654 − 0.755i)4-s + (−0.142 + 0.989i)5-s + (−0.152 − 1.06i)6-s + (−1.45 + 3.19i)7-s + (0.959 − 0.281i)8-s + (−0.768 + 1.68i)9-s + (−0.841 − 0.540i)10-s + (−0.0393 + 0.273i)11-s + (1.02 + 0.302i)12-s + (5.23 + 1.53i)13-s + (−2.30 − 2.65i)14-s + (−0.445 − 0.975i)15-s + (−0.142 + 0.989i)16-s + (−0.879 + 1.01i)17-s + ⋯ |
L(s) = 1 | + (−0.293 + 0.643i)2-s + (−0.520 + 0.334i)3-s + (−0.327 − 0.377i)4-s + (−0.0636 + 0.442i)5-s + (−0.0623 − 0.433i)6-s + (−0.551 + 1.20i)7-s + (0.339 − 0.0996i)8-s + (−0.256 + 0.560i)9-s + (−0.266 − 0.170i)10-s + (−0.0118 + 0.0825i)11-s + (0.297 + 0.0872i)12-s + (1.45 + 0.426i)13-s + (−0.614 − 0.709i)14-s + (−0.115 − 0.251i)15-s + (−0.0355 + 0.247i)16-s + (−0.213 + 0.246i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.164789 - 0.498368i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.164789 - 0.498368i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 5 | \( 1 + (0.142 - 0.989i)T \) |
| 67 | \( 1 + (-8.07 - 1.34i)T \) |
good | 3 | \( 1 + (0.902 - 0.579i)T + (1.24 - 2.72i)T^{2} \) |
| 7 | \( 1 + (1.45 - 3.19i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (0.0393 - 0.273i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-5.23 - 1.53i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.879 - 1.01i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (2.08 + 4.56i)T + (-12.4 + 14.3i)T^{2} \) |
| 23 | \( 1 + (0.354 - 0.227i)T + (9.55 - 20.9i)T^{2} \) |
| 29 | \( 1 + 3.45T + 29T^{2} \) |
| 31 | \( 1 + (8.70 - 2.55i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + 3.03T + 37T^{2} \) |
| 41 | \( 1 + (0.422 - 0.488i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-2.62 + 3.03i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + (4.29 - 2.75i)T + (19.5 - 42.7i)T^{2} \) |
| 53 | \( 1 + (1.14 + 1.32i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-2.95 + 0.867i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-0.749 - 5.21i)T + (-58.5 + 17.1i)T^{2} \) |
| 71 | \( 1 + (10.8 + 12.4i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.28 - 8.95i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-0.0208 - 0.00611i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-1.55 + 10.8i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (2.75 + 1.76i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−11.03147094160780390117137551658, −10.18143158966171382113251365151, −8.984459405577994028207461109536, −8.720797496278003119182616372202, −7.44784764785247348610572403935, −6.38488556568503389222625385359, −5.85359985756220067172082326992, −4.92751232581832354631459425163, −3.60532217573852204764309915834, −2.14379994780510246294424018722,
0.33931682528577711051747941617, 1.49520870073968445691424883090, 3.47291458263861193779860116469, 3.98635068647752478939191843122, 5.53631228973040101457479217068, 6.40571897383445217231946640523, 7.38962780657836397286655582231, 8.375945983577630664885127993411, 9.215325727498912279813916264853, 10.13511979421983848081155629201