L(s) = 1 | + (1.21 + 0.879i)2-s + (−0.309 + 0.951i)3-s + (0.0742 + 0.228i)4-s + (2.08 − 1.51i)5-s + (−1.21 + 0.879i)6-s + (−0.780 − 2.40i)7-s + (0.813 − 2.50i)8-s + (−0.809 − 0.587i)9-s + 3.85·10-s + (3.27 − 0.499i)11-s − 0.240·12-s + (0.809 + 0.587i)13-s + (1.16 − 3.59i)14-s + (0.796 + 2.45i)15-s + (3.57 − 2.59i)16-s + (−0.636 + 0.462i)17-s + ⋯ |
L(s) = 1 | + (0.856 + 0.622i)2-s + (−0.178 + 0.549i)3-s + (0.0371 + 0.114i)4-s + (0.932 − 0.677i)5-s + (−0.494 + 0.359i)6-s + (−0.295 − 0.908i)7-s + (0.287 − 0.885i)8-s + (−0.269 − 0.195i)9-s + 1.21·10-s + (0.988 − 0.150i)11-s − 0.0693·12-s + (0.224 + 0.163i)13-s + (0.312 − 0.961i)14-s + (0.205 + 0.632i)15-s + (0.894 − 0.649i)16-s + (−0.154 + 0.112i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.174i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.25755 + 0.197997i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.25755 + 0.197997i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (-3.27 + 0.499i)T \) |
| 13 | \( 1 + (-0.809 - 0.587i)T \) |
good | 2 | \( 1 + (-1.21 - 0.879i)T + (0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (-2.08 + 1.51i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (0.780 + 2.40i)T + (-5.66 + 4.11i)T^{2} \) |
| 17 | \( 1 + (0.636 - 0.462i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.94 - 5.99i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 4.00T + 23T^{2} \) |
| 29 | \( 1 + (-1.37 - 4.24i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (5.52 + 4.01i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.79 - 5.52i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.313 - 0.966i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 6.60T + 43T^{2} \) |
| 47 | \( 1 + (0.938 - 2.88i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-4.00 - 2.90i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.304 + 0.938i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-7.67 + 5.57i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 1.72T + 67T^{2} \) |
| 71 | \( 1 + (5.68 - 4.12i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.33 - 7.19i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (3.55 + 2.58i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (11.6 - 8.46i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + (8.68 + 6.30i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.12053996970784255471389774322, −10.03213034219846508673040937892, −9.653554029684156211755228298041, −8.528291628788643320519598111509, −7.09937017225342060194856181611, −6.12870791775337165592104550625, −5.58842589334530327424399878554, −4.33632064919941586165522896227, −3.76918180955949196305338542369, −1.37494473303161638727798214457,
2.04278025727522896802144109606, 2.75175043686064128150207672145, 4.11827162165329685621091420598, 5.48930194075778659635217504727, 6.19480149879424626171448824304, 7.14593873832485135264162259767, 8.559699462290816788686352359235, 9.354923609861794628679423893622, 10.55254397301294757214003168398, 11.39233263293222397708148839445