L(s) = 1 | + (−2.77 + 1.60i)3-s + 2.20i·7-s + (3.63 − 6.29i)9-s − 1.20·11-s + (0.866 + 0.5i)13-s + (1.85 − 1.07i)17-s + (−4.30 + 0.673i)19-s + (−3.53 − 6.11i)21-s + (8.02 + 4.63i)23-s + 13.6i·27-s + (4.16 − 7.21i)29-s + 8.26·31-s + (3.34 − 1.92i)33-s + 2.20i·37-s − 3.20·39-s + ⋯ |
L(s) = 1 | + (−1.60 + 0.925i)3-s + 0.833i·7-s + (1.21 − 2.09i)9-s − 0.363·11-s + (0.240 + 0.138i)13-s + (0.449 − 0.259i)17-s + (−0.988 + 0.154i)19-s + (−0.770 − 1.33i)21-s + (1.67 + 0.966i)23-s + 2.63i·27-s + (0.773 − 1.33i)29-s + 1.48·31-s + (0.581 − 0.335i)33-s + 0.362i·37-s − 0.513·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8614794569\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8614794569\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (4.30 - 0.673i)T \) |
good | 3 | \( 1 + (2.77 - 1.60i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 2.20iT - 7T^{2} \) |
| 11 | \( 1 + 1.20T + 11T^{2} \) |
| 13 | \( 1 + (-0.866 - 0.5i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.85 + 1.07i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-8.02 - 4.63i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.16 + 7.21i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 8.26T + 31T^{2} \) |
| 37 | \( 1 - 2.20iT - 37T^{2} \) |
| 41 | \( 1 + (-3.30 - 5.72i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.03 + 1.17i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (7.21 + 4.16i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.232 + 0.134i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.16 + 7.21i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.70 + 2.95i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.84 + 1.06i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.23 - 9.06i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.20 - 2.42i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.20 - 14.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 5.73iT - 83T^{2} \) |
| 89 | \( 1 + (5.40 - 9.36i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (13.6 - 7.87i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.732993267931497083527643774804, −8.877710570998229686244908113848, −7.942297247467875527438258778324, −6.65696450728217884624885665237, −6.21396130516429690277215880725, −5.31358623340445315478700131217, −4.84363391354979038229761989529, −3.90287681403895726409560687569, −2.71654328277695704338005612130, −0.983756175107212814491582511130,
0.53778917212391410770072805186, 1.40428581528097337730112197510, 2.83393314859157944471322213517, 4.38484840411867799103055346158, 4.96055443284269621080108634422, 5.91371528965572365362065274076, 6.63027778603366450533671225479, 7.13181988366088581277515450118, 7.939378678854435059432289288864, 8.821718951254702759411881315764