L(s) = 1 | + 3-s + 5-s + (0.440 + 0.763i)7-s + 9-s + (1.63 + 2.83i)11-s + (0.784 − 1.35i)13-s + 15-s + (−3.66 + 6.35i)17-s + (4.31 − 7.46i)19-s + (0.440 + 0.763i)21-s + (−1.92 + 3.32i)23-s + 25-s + 27-s + (−0.0681 − 0.118i)29-s + (−0.841 − 1.45i)31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + (0.166 + 0.288i)7-s + 0.333·9-s + (0.493 + 0.854i)11-s + (0.217 − 0.376i)13-s + 0.258·15-s + (−0.889 + 1.54i)17-s + (0.989 − 1.71i)19-s + (0.0961 + 0.166i)21-s + (−0.400 + 0.694i)23-s + 0.200·25-s + 0.192·27-s + (−0.0126 − 0.0219i)29-s + (−0.151 − 0.261i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.766563654\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.766563654\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (7.65 - 2.88i)T \) |
good | 7 | \( 1 + (-0.440 - 0.763i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.63 - 2.83i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.784 + 1.35i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.66 - 6.35i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.31 + 7.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.92 - 3.32i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0681 + 0.118i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.841 + 1.45i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.66 - 2.88i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.87 - 4.98i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + (-5.00 - 8.66i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 1.34T + 53T^{2} \) |
| 59 | \( 1 - 2.31T + 59T^{2} \) |
| 61 | \( 1 + (2.41 - 4.18i)T + (-30.5 - 52.8i)T^{2} \) |
| 71 | \( 1 + (4.49 + 7.79i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.99 - 6.91i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.710 + 1.23i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.575 - 0.996i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 7.60T + 89T^{2} \) |
| 97 | \( 1 + (-1.95 + 3.37i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.743391763513480416899556393973, −7.74987587285272790350091126373, −7.19093203640014938349621202325, −6.32778059854314770289060333539, −5.63885702022846367737118194897, −4.63468971378026397698735314248, −4.02179830847671404936434860152, −2.92113752919610766061009149446, −2.15021850169171362069797114235, −1.21845514504784185070885805422,
0.797481871910535157135068308679, 1.90045077095230565948785585028, 2.81980426048259120065089011873, 3.73752139481972614773868959659, 4.40632952300449304896255815018, 5.49304461424655763585628509841, 6.08520448863893821805312718038, 7.05009283372167345433910995665, 7.55777286675039724048070682732, 8.486538052113449155642139525546