L(s) = 1 | + (−0.215 − 2.22i)5-s + (2.32 − 2.32i)7-s + 5.57·11-s + (1.79 − 1.79i)13-s + (−5.56 − 5.56i)17-s + 2.61·19-s + (−4.44 + 4.44i)23-s + (−4.90 + 0.958i)25-s + 2.12i·29-s + 4.52·31-s + (−5.66 − 4.66i)35-s + (5.02 + 5.02i)37-s + 1.73i·41-s + (−1.19 + 1.19i)43-s + (−0.849 − 0.849i)47-s + ⋯ |
L(s) = 1 | + (−0.0963 − 0.995i)5-s + (0.877 − 0.877i)7-s + 1.68·11-s + (0.497 − 0.497i)13-s + (−1.34 − 1.34i)17-s + 0.600·19-s + (−0.927 + 0.927i)23-s + (−0.981 + 0.191i)25-s + 0.394i·29-s + 0.812·31-s + (−0.957 − 0.788i)35-s + (0.826 + 0.826i)37-s + 0.270i·41-s + (−0.182 + 0.182i)43-s + (−0.123 − 0.123i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0450 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0450 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.914788035\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.914788035\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.215 + 2.22i)T \) |
good | 7 | \( 1 + (-2.32 + 2.32i)T - 7iT^{2} \) |
| 11 | \( 1 - 5.57T + 11T^{2} \) |
| 13 | \( 1 + (-1.79 + 1.79i)T - 13iT^{2} \) |
| 17 | \( 1 + (5.56 + 5.56i)T + 17iT^{2} \) |
| 19 | \( 1 - 2.61T + 19T^{2} \) |
| 23 | \( 1 + (4.44 - 4.44i)T - 23iT^{2} \) |
| 29 | \( 1 - 2.12iT - 29T^{2} \) |
| 31 | \( 1 - 4.52T + 31T^{2} \) |
| 37 | \( 1 + (-5.02 - 5.02i)T + 37iT^{2} \) |
| 41 | \( 1 - 1.73iT - 41T^{2} \) |
| 43 | \( 1 + (1.19 - 1.19i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.849 + 0.849i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.22 + 4.22i)T + 53iT^{2} \) |
| 59 | \( 1 + 8.08iT - 59T^{2} \) |
| 61 | \( 1 + 3.13iT - 61T^{2} \) |
| 67 | \( 1 + (-1.86 - 1.86i)T + 67iT^{2} \) |
| 71 | \( 1 + 6.95iT - 71T^{2} \) |
| 73 | \( 1 + (5.86 + 5.86i)T + 73iT^{2} \) |
| 79 | \( 1 - 2.52iT - 79T^{2} \) |
| 83 | \( 1 + (-0.694 - 0.694i)T + 83iT^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + (-4.09 + 4.09i)T - 97iT^{2} \) |
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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
−9.347149600775755598139800580484, −8.519345658459552751921265564038, −7.83084755060046122380093227488, −6.95883732820628001450864681731, −6.06501766140331939217680375587, −4.87270430312363090668323973362, −4.39950771348534735955401002882, −3.44576624930402248552647977406, −1.70389885265179269173244404955, −0.844980212121450216445762351068,
1.62783913139476686946615507829, 2.48536455810687830817378652029, 3.90034989090359303277807452581, 4.39632741604245209611747429404, 5.98233563361629350642587423390, 6.29396221369426254967053281386, 7.20037702210954690877111790068, 8.330334786063456668846835399191, 8.770321599481731210856598478867, 9.674262946948898347861401957609