| L(s) = 1 | + (−0.573 + 0.819i)2-s + (−0.342 − 0.939i)4-s + (−0.718 + 0.0628i)5-s + (−2.07 − 1.73i)7-s + (0.965 + 0.258i)8-s + (0.360 − 0.624i)10-s + (2.14 + 3.71i)11-s + (0.0350 + 0.0163i)13-s + (2.61 − 0.699i)14-s + (−0.766 + 0.642i)16-s + (−1.26 + 0.588i)17-s + (−1.15 + 0.812i)19-s + (0.304 + 0.653i)20-s + (−4.27 − 0.374i)22-s + (1.21 + 4.54i)23-s + ⋯ |
| L(s) = 1 | + (−0.405 + 0.579i)2-s + (−0.171 − 0.469i)4-s + (−0.321 + 0.0280i)5-s + (−0.782 − 0.656i)7-s + (0.341 + 0.0915i)8-s + (0.113 − 0.197i)10-s + (0.647 + 1.12i)11-s + (0.00973 + 0.00453i)13-s + (0.697 − 0.186i)14-s + (−0.191 + 0.160i)16-s + (−0.306 + 0.142i)17-s + (−0.266 + 0.186i)19-s + (0.0681 + 0.146i)20-s + (−0.911 − 0.0797i)22-s + (0.253 + 0.947i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.191236 + 0.571634i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.191236 + 0.571634i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.573 - 0.819i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (-6.02 - 0.856i)T \) |
| good | 5 | \( 1 + (0.718 - 0.0628i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (2.07 + 1.73i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-2.14 - 3.71i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.0350 - 0.0163i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (1.26 - 0.588i)T + (10.9 - 13.0i)T^{2} \) |
| 19 | \( 1 + (1.15 - 0.812i)T + (6.49 - 17.8i)T^{2} \) |
| 23 | \( 1 + (-1.21 - 4.54i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (2.18 - 8.15i)T + (-25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (3.19 - 3.19i)T - 31iT^{2} \) |
| 41 | \( 1 + (8.42 - 3.06i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.673 + 0.673i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.04 - 1.18i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.385 + 0.459i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-0.273 + 3.12i)T + (-58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (4.73 - 10.1i)T + (-39.2 - 46.7i)T^{2} \) |
| 67 | \( 1 + (-5.48 + 6.53i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.759 - 0.133i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 - 9.36iT - 73T^{2} \) |
| 79 | \( 1 + (0.124 + 1.41i)T + (-77.7 + 13.7i)T^{2} \) |
| 83 | \( 1 + (2.84 - 7.81i)T + (-63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (10.6 + 0.934i)T + (87.6 + 15.4i)T^{2} \) |
| 97 | \( 1 + (2.87 - 0.769i)T + (84.0 - 48.5i)T^{2} \) |
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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
−10.65992589661026394961089019402, −9.792223866362278012880866944910, −9.256850687995611494293304863361, −8.170145939170588477651221824240, −7.12408606401859636645250098925, −6.81527684877758853995204698386, −5.58442732986120745510027036413, −4.39652769923477932153708433523, −3.45872786595843084974100041021, −1.59296795945750946581232692358,
0.37013033472968230052093994492, 2.26496537525727920077386230809, 3.35865717576444763178224911109, 4.30055058057090865988746054157, 5.81106868250249339657426626622, 6.53806300646348558735488404985, 7.78957392239352592706749100533, 8.658394967169801662284209003462, 9.288835530146546485014084467335, 10.10417864712527261371225338492
