| L(s) = 1 | + (−0.951 + 0.309i)2-s + (−1.26 − 1.18i)3-s + (0.809 − 0.587i)4-s + (−0.866 + 0.5i)5-s + (1.56 + 0.739i)6-s + (0.426 + 4.05i)7-s + (−0.587 + 0.809i)8-s + (0.182 + 2.99i)9-s + (0.669 − 0.743i)10-s + (−3.30 − 1.47i)11-s + (−1.71 − 0.218i)12-s + (−0.315 − 1.48i)13-s + (−1.66 − 3.72i)14-s + (1.68 + 0.397i)15-s + (0.309 − 0.951i)16-s + (0.831 − 0.370i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (−0.728 − 0.685i)3-s + (0.404 − 0.293i)4-s + (−0.387 + 0.223i)5-s + (0.639 + 0.301i)6-s + (0.161 + 1.53i)7-s + (−0.207 + 0.286i)8-s + (0.0608 + 0.998i)9-s + (0.211 − 0.235i)10-s + (−0.996 − 0.443i)11-s + (−0.495 − 0.0631i)12-s + (−0.0875 − 0.411i)13-s + (−0.443 − 0.996i)14-s + (0.435 + 0.102i)15-s + (0.0772 − 0.237i)16-s + (0.201 − 0.0897i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.152250 - 0.240244i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.152250 - 0.240244i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (1.26 + 1.18i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 31 | \( 1 + (-3.18 + 4.56i)T \) |
| good | 7 | \( 1 + (-0.426 - 4.05i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (3.30 + 1.47i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (0.315 + 1.48i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-0.831 + 0.370i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (5.99 + 1.27i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-5.09 - 3.70i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.234 + 0.722i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-6.51 - 3.76i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.77 + 4.30i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-2.06 + 9.73i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (9.32 + 3.03i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.09 + 10.4i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (6.39 - 5.75i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 10.8iT - 61T^{2} \) |
| 67 | \( 1 + (3.17 + 5.50i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.69 + 0.283i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (4.41 - 9.91i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (5.32 + 11.9i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (4.85 - 5.39i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (5.08 - 3.69i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-10.0 + 7.32i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.841390917234103064434224697670, −8.637437776254622428022472424210, −8.207861019050604866229436330322, −7.33448569764645545050667161741, −6.36721549424352307551103485925, −5.63058189957936196727181289815, −4.91889283168836868352867631700, −2.92180458856522522075810805503, −2.01850709218946781776073884749, −0.20454359620905476453825853767,
1.16532161268065548662008532621, 3.03335223944424006972829303004, 4.35118657172079965672272190995, 4.67467190931401747049851108026, 6.20431497137128481377755142488, 7.04566097512858519754007854597, 7.83452864233899166602238804984, 8.743082721488151813838337339212, 9.772942000249987896447248938826, 10.45974340629423890777605036487
