L(s) = 1 | + (1.43 + 2.48i)3-s − i·5-s + (0.0113 + 0.00658i)7-s + (−2.62 + 4.53i)9-s + (0.541 − 0.312i)11-s + (1.97 + 3.01i)13-s + (2.48 − 1.43i)15-s + (−3.78 + 6.54i)17-s + (6.29 + 3.63i)19-s + 0.0377i·21-s + (−1.22 − 2.13i)23-s − 25-s − 6.43·27-s + (−1.15 − 2.00i)29-s − 8.02i·31-s + ⋯ |
L(s) = 1 | + (0.828 + 1.43i)3-s − 0.447i·5-s + (0.00430 + 0.00248i)7-s + (−0.873 + 1.51i)9-s + (0.163 − 0.0942i)11-s + (0.547 + 0.837i)13-s + (0.641 − 0.370i)15-s + (−0.916 + 1.58i)17-s + (1.44 + 0.833i)19-s + 0.00824i·21-s + (−0.256 − 0.444i)23-s − 0.200·25-s − 1.23·27-s + (−0.215 − 0.373i)29-s − 1.44i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0423 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0423 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.27476 + 1.32997i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.27476 + 1.32997i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 + (-1.97 - 3.01i)T \) |
good | 3 | \( 1 + (-1.43 - 2.48i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-0.0113 - 0.00658i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.541 + 0.312i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (3.78 - 6.54i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.29 - 3.63i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.22 + 2.13i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.15 + 2.00i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8.02iT - 31T^{2} \) |
| 37 | \( 1 + (3.65 - 2.10i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.29 + 1.90i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.39 + 2.42i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 4.00iT - 47T^{2} \) |
| 53 | \( 1 - 6.25T + 53T^{2} \) |
| 59 | \( 1 + (11.6 + 6.74i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.28 + 5.69i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.81 + 3.35i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-13.2 - 7.63i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 2.98iT - 73T^{2} \) |
| 79 | \( 1 - 2.02T + 79T^{2} \) |
| 83 | \( 1 - 2.35iT - 83T^{2} \) |
| 89 | \( 1 + (-4.50 + 2.59i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.02 - 4.05i)T + (48.5 + 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.91608111714844983597290795421, −10.02258214424741167060964178913, −9.360182856364490902857829348782, −8.601220470102421697796618355305, −7.938164288257196839904487461016, −6.36942952710762430213058178919, −5.26274782730777445691905665210, −4.10017184155106943706403941508, −3.67139889096209086959803556391, −2.03161299245313948593089133296,
1.08947827079650667427459457065, 2.58287055536551451700745782435, 3.30438893654510820993026081147, 5.10636010537989316240223784707, 6.37629698877426067637402314345, 7.22634579593162771291411724651, 7.67287347118950559851900630065, 8.824783298525500995087163135919, 9.441979519967381795353415795180, 10.81637136022381080880626060398