L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.866 + 0.5i)3-s − 1.00i·4-s + (−0.0285 + 0.106i)5-s + (−0.258 + 0.965i)6-s + (0.253 + 2.63i)7-s + (−0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (0.0551 + 0.0955i)10-s + (−1.22 + 4.58i)11-s + (0.500 + 0.866i)12-s + (3.58 − 0.403i)13-s + (2.04 + 1.68i)14-s + (−0.0285 − 0.106i)15-s − 1.00·16-s + 2.90·17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.499 + 0.288i)3-s − 0.500i·4-s + (−0.0127 + 0.0476i)5-s + (−0.105 + 0.394i)6-s + (0.0958 + 0.995i)7-s + (−0.250 − 0.250i)8-s + (0.166 − 0.288i)9-s + (0.0174 + 0.0302i)10-s + (−0.370 + 1.38i)11-s + (0.144 + 0.250i)12-s + (0.993 − 0.111i)13-s + (0.545 + 0.449i)14-s + (−0.00737 − 0.0275i)15-s − 0.250·16-s + 0.704·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.49287 + 0.427771i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49287 + 0.427771i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (-0.253 - 2.63i)T \) |
| 13 | \( 1 + (-3.58 + 0.403i)T \) |
good | 5 | \( 1 + (0.0285 - 0.106i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (1.22 - 4.58i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 - 2.90T + 17T^{2} \) |
| 19 | \( 1 + (2.58 - 0.692i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 - 5.34iT - 23T^{2} \) |
| 29 | \( 1 + (0.975 - 1.68i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-9.39 + 2.51i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (5.39 + 5.39i)T + 37iT^{2} \) |
| 41 | \( 1 + (-6.25 + 1.67i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (2.18 - 1.26i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.89 + 1.58i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (3.68 - 6.38i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.23 + 2.23i)T - 59iT^{2} \) |
| 61 | \( 1 + (-6.85 - 3.95i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.83 - 2.10i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (5.73 + 1.53i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (1.38 + 5.15i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (1.92 + 3.32i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.03 + 8.03i)T + 83iT^{2} \) |
| 89 | \( 1 + (10.0 - 10.0i)T - 89iT^{2} \) |
| 97 | \( 1 + (-3.07 + 11.4i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−11.00272197823550106955001458621, −10.11422875864717897335424786832, −9.393775195333797195772391782523, −8.341709721028194777429288547302, −7.09573906285261958406857540647, −5.97131770004790497182259087474, −5.25983332129941602569588858869, −4.28237426237314004816767485271, −3.03305433296196821031033297806, −1.66622126322335749762963187065,
0.892752710612980708793149787857, 3.03162598931001793629809880932, 4.16958778687495746949248819860, 5.17954418408487680144371365787, 6.30975862614258639812114802669, 6.75131917716965573790745742569, 8.166020548088731291241645309349, 8.430769300294879246381945268219, 10.10814105452427809608033214906, 10.86171729993332210587170169132