L(s) = 1 | + (−0.939 + 0.342i)3-s + (−0.0537 + 0.304i)5-s + (−4.22 + 2.43i)7-s + (0.766 − 0.642i)9-s + (−1.67 − 0.966i)11-s + (1.49 − 4.11i)13-s + (−0.0537 − 0.304i)15-s + (−3.57 − 2.99i)17-s + (3.42 − 2.69i)19-s + (3.13 − 3.73i)21-s + (7.72 − 1.36i)23-s + (4.60 + 1.67i)25-s + (−0.500 + 0.866i)27-s + (5.51 + 6.57i)29-s + (1.67 + 2.89i)31-s + ⋯ |
L(s) = 1 | + (−0.542 + 0.197i)3-s + (−0.0240 + 0.136i)5-s + (−1.59 + 0.922i)7-s + (0.255 − 0.214i)9-s + (−0.504 − 0.291i)11-s + (0.415 − 1.14i)13-s + (−0.0138 − 0.0786i)15-s + (−0.866 − 0.727i)17-s + (0.786 − 0.617i)19-s + (0.684 − 0.815i)21-s + (1.61 − 0.283i)23-s + (0.921 + 0.335i)25-s + (−0.0962 + 0.166i)27-s + (1.02 + 1.22i)29-s + (0.300 + 0.519i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.812494 - 0.310692i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.812494 - 0.310692i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.939 - 0.342i)T \) |
| 19 | \( 1 + (-3.42 + 2.69i)T \) |
good | 5 | \( 1 + (0.0537 - 0.304i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (4.22 - 2.43i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.67 + 0.966i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.49 + 4.11i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (3.57 + 2.99i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-7.72 + 1.36i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.51 - 6.57i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.67 - 2.89i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.17iT - 37T^{2} \) |
| 41 | \( 1 + (3.68 + 10.1i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (2.64 + 0.465i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (5.72 + 6.82i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-11.3 + 2.00i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-2.80 - 2.35i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.0154 + 0.0878i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (3.50 - 2.93i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.43 + 13.8i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (11.0 - 4.02i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (4.56 - 1.66i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-9.83 + 5.67i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.29 + 11.7i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (10.0 - 11.9i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−10.18556133377100673869660161366, −8.992252568479473457459156271593, −8.737729886882930976261786338937, −7.04235138531861936031864899262, −6.68722714866545397846094563130, −5.49630675007486389300944085015, −5.03388081539103561754864227736, −3.22651344805255981960069894605, −2.87782706312635644792430712989, −0.54394667566891387571857105763,
1.06736602419780731139988290609, 2.80100207611639607274060638832, 3.95575336839438779442175998822, 4.80343356397948956230903667281, 6.19877251572031980793386091622, 6.62974468689679915186456334018, 7.40819818657310302503548020605, 8.566503490993778822769271915345, 9.577770805128388262942926417857, 10.12812575714602017344348907080