L(s) = 1 | + (−1.04 + 1.38i)3-s − 0.0619·5-s + (−1.63 + 2.07i)7-s + (−0.828 − 2.88i)9-s − 3.18·11-s + (−0.252 − 0.437i)13-s + (0.0645 − 0.0857i)15-s + (−0.554 − 0.960i)17-s + (0.933 − 1.61i)19-s + (−1.16 − 4.43i)21-s − 6.20·23-s − 4.99·25-s + (4.85 + 1.85i)27-s + (2.39 − 4.15i)29-s + (1.26 − 2.19i)31-s + ⋯ |
L(s) = 1 | + (−0.601 + 0.798i)3-s − 0.0277·5-s + (−0.618 + 0.785i)7-s + (−0.276 − 0.961i)9-s − 0.958·11-s + (−0.0700 − 0.121i)13-s + (0.0166 − 0.0221i)15-s + (−0.134 − 0.233i)17-s + (0.214 − 0.370i)19-s + (−0.255 − 0.966i)21-s − 1.29·23-s − 0.999·25-s + (0.933 + 0.357i)27-s + (0.445 − 0.770i)29-s + (0.227 − 0.394i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 + 0.521i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0305179 - 0.108444i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0305179 - 0.108444i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.04 - 1.38i)T \) |
| 7 | \( 1 + (1.63 - 2.07i)T \) |
good | 5 | \( 1 + 0.0619T + 5T^{2} \) |
| 11 | \( 1 + 3.18T + 11T^{2} \) |
| 13 | \( 1 + (0.252 + 0.437i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.554 + 0.960i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.933 + 1.61i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 6.20T + 23T^{2} \) |
| 29 | \( 1 + (-2.39 + 4.15i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.26 + 2.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.26 - 7.38i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.94 + 8.56i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.95 - 6.85i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.29 + 5.70i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.58 + 2.74i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.50 - 7.80i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.94 - 12.0i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.66 - 2.88i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2.25T + 71T^{2} \) |
| 73 | \( 1 + (-2.07 - 3.59i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.48 - 2.57i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.17 + 3.76i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (4.30 - 7.44i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.27 - 5.67i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.65189443723025222733212939802, −10.23295029040486172516781855616, −9.984812242873323205593241195895, −8.910845954375914723867339008055, −7.978667459731989141719670116011, −6.61429872839689419818082211801, −5.75801531765672803652117889362, −4.97418448480802350585478024563, −3.73330259203713821202953283256, −2.53167665062600038268030748476,
0.06760020064369572246790728837, 1.86694089124005907912493350206, 3.40036707908103213339346621024, 4.78706272033078112926141454985, 5.87304303609717554391468694159, 6.70682943442101941329267473511, 7.61745162224314768961449254733, 8.291636952179889327579248266946, 9.786054068284845181187238161574, 10.44886870958490888858031545426