L(s) = 1 | + (1.69 + 0.377i)3-s + (1.05 + 0.282i)5-s + (1.93 − 3.35i)7-s + (2.71 + 1.27i)9-s + (−3.53 + 0.946i)11-s + (3.87 + 1.03i)13-s + (1.67 + 0.874i)15-s + 1.55i·17-s + (−4.06 − 4.06i)19-s + (4.53 − 4.93i)21-s + (3.86 − 2.23i)23-s + (−3.29 − 1.90i)25-s + (4.10 + 3.18i)27-s + (−4.28 + 1.14i)29-s + (−1.85 + 1.07i)31-s + ⋯ |
L(s) = 1 | + (0.975 + 0.217i)3-s + (0.471 + 0.126i)5-s + (0.731 − 1.26i)7-s + (0.905 + 0.425i)9-s + (−1.06 + 0.285i)11-s + (1.07 + 0.288i)13-s + (0.432 + 0.225i)15-s + 0.375i·17-s + (−0.931 − 0.931i)19-s + (0.990 − 1.07i)21-s + (0.806 − 0.465i)23-s + (−0.659 − 0.380i)25-s + (0.790 + 0.612i)27-s + (−0.795 + 0.213i)29-s + (−0.333 + 0.192i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.29499 - 0.0944635i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.29499 - 0.0944635i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.69 - 0.377i)T \) |
good | 5 | \( 1 + (-1.05 - 0.282i)T + (4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (-1.93 + 3.35i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.53 - 0.946i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-3.87 - 1.03i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 - 1.55iT - 17T^{2} \) |
| 19 | \( 1 + (4.06 + 4.06i)T + 19iT^{2} \) |
| 23 | \( 1 + (-3.86 + 2.23i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.28 - 1.14i)T + (25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (1.85 - 1.07i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.04 - 6.04i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1.59 - 2.76i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.47 - 5.51i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (0.0494 - 0.0856i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.72 - 1.72i)T - 53iT^{2} \) |
| 59 | \( 1 + (3.58 - 13.3i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (2.33 + 8.69i)T + (-52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (0.251 - 0.939i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 7.11iT - 71T^{2} \) |
| 73 | \( 1 + 10.4iT - 73T^{2} \) |
| 79 | \( 1 + (14.2 + 8.22i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.02 - 15.0i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + (-0.0532 + 0.0922i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.74155184520215229863936237407, −9.888261661727479512809319177277, −8.894433357057450067244510687483, −8.054081451072236173041928800672, −7.36663668172337722098720781016, −6.30607064005306081029523876479, −4.79085820642394455340856014718, −4.10764404390428384080911081416, −2.78940707028909303430964227388, −1.52687744019216850049418052795,
1.74115638192538939245911542365, 2.63506222923115662941870608004, 3.89471382590944192258903191600, 5.37651306714682909099054018675, 5.96624291143049457252352587323, 7.45925435667685792414611717093, 8.241860223810871184236764223876, 8.840139423968076084990171890710, 9.627409194995185723246475108033, 10.72630255432782354266974329697