L(s) = 1 | + (−0.666 + 1.59i)3-s − 0.468·5-s + (2.39 − 1.13i)7-s + (−2.11 − 2.12i)9-s + 1.34·11-s + (−3.16 − 5.48i)13-s + (0.311 − 0.748i)15-s + (−2.47 − 4.28i)17-s + (−2.38 + 4.13i)19-s + (0.222 + 4.57i)21-s − 7.62·23-s − 4.78·25-s + (4.81 − 1.95i)27-s + (−1.80 + 3.12i)29-s + (3.24 − 5.62i)31-s + ⋯ |
L(s) = 1 | + (−0.384 + 0.923i)3-s − 0.209·5-s + (0.903 − 0.428i)7-s + (−0.704 − 0.709i)9-s + 0.406·11-s + (−0.877 − 1.52i)13-s + (0.0805 − 0.193i)15-s + (−0.599 − 1.03i)17-s + (−0.548 + 0.949i)19-s + (0.0485 + 0.998i)21-s − 1.59·23-s − 0.956·25-s + (0.926 − 0.377i)27-s + (−0.335 + 0.580i)29-s + (0.583 − 1.01i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0662 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0662 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7691194539\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7691194539\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.666 - 1.59i)T \) |
| 7 | \( 1 + (-2.39 + 1.13i)T \) |
good | 5 | \( 1 + 0.468T + 5T^{2} \) |
| 11 | \( 1 - 1.34T + 11T^{2} \) |
| 13 | \( 1 + (3.16 + 5.48i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.47 + 4.28i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.38 - 4.13i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 7.62T + 23T^{2} \) |
| 29 | \( 1 + (1.80 - 3.12i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.24 + 5.62i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.24 + 9.07i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0251 + 0.0435i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.431 + 0.748i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.49 + 9.51i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.84 - 10.1i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.93 - 3.35i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.87 + 3.24i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.32 - 2.29i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 7.04T + 71T^{2} \) |
| 73 | \( 1 + (3.30 + 5.71i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.58 - 2.75i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.90 - 8.49i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.30 + 9.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.97 + 12.0i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.02923484734783451032715722256, −9.044276653603055659052650886679, −8.021942674416442412424250216830, −7.48153138770325653017455905122, −6.08331788714062191027060205647, −5.36972256939536584090099131667, −4.41296576963827500950941043669, −3.73795577919408682911822996393, −2.30887668444107881720974440470, −0.35374302637040764173362015395,
1.66842241916251186375424545845, 2.35052197481498143259404866749, 4.19678148560598004552742706418, 4.88523353658264798961293109571, 6.18000587905710334842521679701, 6.63550851190161932486957783852, 7.74218206627021266133636470120, 8.317239557678642252472061688845, 9.176224080108755648870095835463, 10.23465328958691905340160982855