L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s − 4.37·5-s + 0.999·8-s + (2.18 + 3.78i)10-s + 1.37·11-s + (−1 − 1.73i)13-s + (−0.5 − 0.866i)16-s + (0.686 + 1.18i)17-s + (−2.5 + 4.33i)19-s + (2.18 − 3.78i)20-s + (−0.686 − 1.18i)22-s + 1.62·23-s + 14.1·25-s + (−0.999 + 1.73i)26-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s − 1.95·5-s + 0.353·8-s + (0.691 + 1.19i)10-s + 0.413·11-s + (−0.277 − 0.480i)13-s + (−0.125 − 0.216i)16-s + (0.166 + 0.288i)17-s + (−0.573 + 0.993i)19-s + (0.488 − 0.846i)20-s + (−0.146 − 0.253i)22-s + 0.339·23-s + 2.82·25-s + (−0.196 + 0.339i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4578071674\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4578071674\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 4.37T + 5T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.686 - 1.18i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 1.62T + 23T^{2} \) |
| 29 | \( 1 + (4.37 - 7.57i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.31 - 4.00i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.05 + 7.02i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.37 + 7.57i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.05 - 8.76i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.55 + 2.69i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.05 + 1.83i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 7.11T + 71T^{2} \) |
| 73 | \( 1 + (6.05 + 10.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.55 - 4.43i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.74 + 15.1i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.37 + 12.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.05 + 7.02i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.616851655551677360876459561435, −7.84744558500904813901744223875, −7.42828440304770580599152972627, −6.52909674291594173444756900864, −5.23127497932271184477592250270, −4.34871802716806614813430058164, −3.63116879773292896030957052502, −3.07596571046677670691874317417, −1.55152617679089887053512646638, −0.25346285820341548265606375768,
0.817645971777291304580223704905, 2.55381423426269280673360830379, 3.79336767974122541992016669289, 4.32830698603519802808455166401, 5.11020444346490057931708584325, 6.34596704540668470434794396383, 7.01299130388191165441014999316, 7.68206997588125667594110240582, 8.124808342405388511272713552785, 9.075236700587501965210457816008