L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.448 − 1.67i)5-s + (0.633 + 1.09i)7-s + (−0.707 − 0.707i)8-s + 1.73·10-s + 2.44·11-s + (0.366 + 0.0980i)13-s + (−0.896 + 0.896i)14-s + (0.500 − 0.866i)16-s + (6.24 − 1.67i)17-s + (−5.09 − 1.36i)19-s + (0.448 + 1.67i)20-s + (0.633 + 2.36i)22-s + (2.44 + 2.44i)23-s + ⋯ |
L(s) = 1 | + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (0.200 − 0.748i)5-s + (0.239 + 0.415i)7-s + (−0.249 − 0.249i)8-s + 0.547·10-s + 0.738·11-s + (0.101 + 0.0272i)13-s + (−0.239 + 0.239i)14-s + (0.125 − 0.216i)16-s + (1.51 − 0.405i)17-s + (−1.16 − 0.313i)19-s + (0.100 + 0.374i)20-s + (0.135 + 0.504i)22-s + (0.510 + 0.510i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 - 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.67534 + 0.575152i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.67534 + 0.575152i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (4.69 - 3.86i)T \) |
good | 5 | \( 1 + (-0.448 + 1.67i)T + (-4.33 - 2.5i)T^{2} \) |
| 7 | \( 1 + (-0.633 - 1.09i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + (-0.366 - 0.0980i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (-6.24 + 1.67i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (5.09 + 1.36i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-2.44 - 2.44i)T + 23iT^{2} \) |
| 29 | \( 1 + (-6.12 + 6.12i)T - 29iT^{2} \) |
| 31 | \( 1 + (-5.73 - 5.73i)T + 31iT^{2} \) |
| 41 | \( 1 + (-2.89 - 5.01i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.267 + 0.267i)T - 43iT^{2} \) |
| 47 | \( 1 + 4.24iT - 47T^{2} \) |
| 53 | \( 1 + (5.22 + 3.01i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.79 - 1.55i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (2.86 - 10.6i)T + (-52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (3.63 - 2.09i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.12 + 1.22i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 6.39iT - 73T^{2} \) |
| 79 | \( 1 + (3.36 + 0.901i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (3.10 + 1.79i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.10 + 4.12i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-0.366 + 0.366i)T - 97iT^{2} \) |
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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
−10.43895488853707749489196595453, −9.527308898511276293083118288531, −8.712799635800506539286652719997, −8.142886508196233026545012070251, −6.97835593038259389592726996261, −6.10473265847633147745931890353, −5.16070787412368392995015582497, −4.40971052458637289633532256140, −3.04260584665439368634485774319, −1.23893179478921881525250639556,
1.25301994417148307967651445148, 2.68299399064931606871906600736, 3.71113204577976238502568341556, 4.68069765337814322350518526301, 5.97921633324547839831474576955, 6.74427942327671224693348336928, 7.88755210502602257668803570062, 8.813541445024654687636063276287, 9.836160915613150038540608809125, 10.62610687800635041544920703124