L(s) = 1 | + (1.5 + 0.866i)3-s + (−0.267 + i)5-s + (2.36 + 1.36i)7-s + (1.5 + 2.59i)9-s + (4.23 − 1.13i)11-s + (−3.36 − 0.901i)13-s + (−1.26 + 1.26i)15-s − 5.73·17-s + (2.36 − 2.36i)19-s + (2.36 + 4.09i)21-s + (−4.09 + 2.36i)23-s + (3.40 + 1.96i)25-s + 5.19i·27-s + (−0.633 − 2.36i)29-s + (0.267 + 0.464i)31-s + ⋯ |
L(s) = 1 | + (0.866 + 0.499i)3-s + (−0.119 + 0.447i)5-s + (0.894 + 0.516i)7-s + (0.5 + 0.866i)9-s + (1.27 − 0.341i)11-s + (−0.933 − 0.250i)13-s + (−0.327 + 0.327i)15-s − 1.39·17-s + (0.542 − 0.542i)19-s + (0.516 + 0.894i)21-s + (−0.854 + 0.493i)23-s + (0.680 + 0.392i)25-s + 0.999i·27-s + (−0.117 − 0.439i)29-s + (0.0481 + 0.0833i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.83283 + 1.00552i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.83283 + 1.00552i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.267 - i)T + (-4.33 - 2.5i)T^{2} \) |
| 7 | \( 1 + (-2.36 - 1.36i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.23 + 1.13i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (3.36 + 0.901i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + 5.73T + 17T^{2} \) |
| 19 | \( 1 + (-2.36 + 2.36i)T - 19iT^{2} \) |
| 23 | \( 1 + (4.09 - 2.36i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.633 + 2.36i)T + (-25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (-0.267 - 0.464i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.73 - 4.73i)T + 37iT^{2} \) |
| 41 | \( 1 + (-2.59 + 1.5i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-8.33 + 2.23i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (3.83 - 6.63i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.46 + 7.46i)T + 53iT^{2} \) |
| 59 | \( 1 + (-1.96 + 7.33i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (3 + 11.1i)T + (-52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (6.59 + 1.76i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 2.92iT - 71T^{2} \) |
| 73 | \( 1 - 6.26iT - 73T^{2} \) |
| 79 | \( 1 + (-6 + 10.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.366 + 1.36i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 - 2iT - 89T^{2} \) |
| 97 | \( 1 + (5.86 - 10.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.04601386011717249705916517621, −9.726108973614955566295964446959, −9.179509816783846416751313992950, −8.297115184287792396439669398711, −7.47500259637307731378188261310, −6.44067288560019112536254488185, −5.04513515440582730632390454204, −4.22426791487876113833159390421, −3.01140495584972192352855143522, −1.91493704273533030447488103580,
1.26848366388194224536007203471, 2.40603293811004784736397263174, 4.07698259689205053550646436199, 4.58169251470589773119412884980, 6.26041863570242261577086910155, 7.20392190230465880518958012204, 7.87717361387222809149298613794, 8.881611924631788124660252003312, 9.394310191057611213834787191377, 10.53845161253055456786191031745