L(s) = 1 | + (0.157 + 2.82i)2-s + (−7.95 + 0.888i)4-s + (1.23 − 0.715i)5-s + (−23.8 − 13.7i)7-s + (−3.76 − 22.3i)8-s + (2.21 + 3.38i)10-s + (−11.1 + 19.2i)11-s + (−34.5 − 59.9i)13-s + (35.1 − 69.6i)14-s + (62.4 − 14.1i)16-s + 31.4i·17-s − 11.4i·19-s + (−9.21 + 6.78i)20-s + (−56.0 − 28.3i)22-s + (−72.6 − 125. i)23-s + ⋯ |
L(s) = 1 | + (0.0556 + 0.998i)2-s + (−0.993 + 0.111i)4-s + (0.110 − 0.0639i)5-s + (−1.28 − 0.744i)7-s + (−0.166 − 0.986i)8-s + (0.0700 + 0.107i)10-s + (−0.304 + 0.527i)11-s + (−0.738 − 1.27i)13-s + (0.671 − 1.32i)14-s + (0.975 − 0.220i)16-s + 0.448i·17-s − 0.138i·19-s + (−0.102 + 0.0758i)20-s + (−0.543 − 0.274i)22-s + (−0.658 − 1.14i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.131 + 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.131 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.190179 - 0.217028i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.190179 - 0.217028i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.157 - 2.82i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.23 + 0.715i)T + (62.5 - 108. i)T^{2} \) |
| 7 | \( 1 + (23.8 + 13.7i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (11.1 - 19.2i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (34.5 + 59.9i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 31.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 11.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (72.6 + 125. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-93.6 - 54.0i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-102. + 59.3i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 300.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (344. - 199. i)T + (3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (173. + 100. i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (151. - 262. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 243. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (41.9 + 72.6i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-199. + 345. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-307. + 177. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 866.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 64.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + (354. + 204. i)T + (2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-79.8 + 138. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 1.49e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-700. + 1.21e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.02188769289505187287068971566, −12.44761738121612340857579536417, −10.27753928118727000813453035207, −9.822153650169887624960046862925, −8.327189532647507629103571601392, −7.22981494901524479016281651150, −6.25588308207095518531641743505, −4.88794309829254508587488494321, −3.36256936768755026074541497196, −0.14902179528642485069975522841,
2.28136591066239083552444470907, 3.59427416190026295439793358820, 5.25445497180777568314177267881, 6.60700164991727201804785876578, 8.452609261968922723153605620026, 9.530828351979645655815489616422, 10.15270706698335341716154429589, 11.73304556187499251516824883953, 12.17798478688378572746257334427, 13.44856198211586949383001356639