L(s) = 1 | + (−1.39 + 0.245i)2-s + (1.87 − 0.684i)4-s + (−2.10 − 0.764i)5-s + (−1.61 + 2.80i)7-s + (−2.44 + 1.41i)8-s + (−0.520 + 2.95i)9-s + (3.11 + 0.549i)10-s + (−3.28 − 5.69i)11-s + (0.671 − 0.800i)13-s + (1.56 − 4.29i)14-s + (3.06 − 2.57i)16-s − 4.24i·18-s + (3.63 + 2.40i)19-s − 4.47·20-s + (5.97 + 7.12i)22-s + (8.34 − 3.03i)23-s + ⋯ |
L(s) = 1 | + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.939 − 0.342i)5-s + (−0.611 + 1.05i)7-s + (−0.866 + 0.500i)8-s + (−0.173 + 0.984i)9-s + (0.984 + 0.173i)10-s + (−0.991 − 1.71i)11-s + (0.186 − 0.221i)13-s + (0.418 − 1.14i)14-s + (0.766 − 0.642i)16-s − 0.999i·18-s + (0.833 + 0.551i)19-s − 1.00·20-s + (1.27 + 1.51i)22-s + (1.73 − 0.633i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.578511 - 0.166782i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.578511 - 0.166782i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.39 - 0.245i)T \) |
| 5 | \( 1 + (2.10 + 0.764i)T \) |
| 19 | \( 1 + (-3.63 - 2.40i)T \) |
good | 3 | \( 1 + (0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (1.61 - 2.80i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.28 + 5.69i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.671 + 0.800i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-8.34 + 3.03i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 11.5iT - 37T^{2} \) |
| 41 | \( 1 + (2.67 + 3.19i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.59 + 9.01i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-3.38 - 9.30i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-15.0 + 2.65i)T + (55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.444 - 0.529i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-91.1 + 33.1i)T^{2} \) |
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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
−10.37747159907446950400899738297, −9.007059681423251481127403656486, −8.631334438574677641868404323373, −7.913447339938244412349691047629, −7.07280697405921569789207911994, −5.67394491881736806433826767132, −5.35091153396362250659705371455, −3.34824427642687433677063181906, −2.55397217699460135112568851123, −0.56423332657929182261745614051,
0.943477625592680960752854654641, 2.83738177937458675012931770150, 3.63013337653311814077063465761, 4.86330545973374251957658733717, 6.64778048580021229086856400500, 7.09373808832035631364275066267, 7.66841674791486568595663491267, 8.780093155950521332854683154726, 9.780770986035567102931055254987, 10.15183671233890860851112649903