L(s) = 1 | + (−5.44 + 5.44i)2-s + (1.36 + 1.36i)3-s − 43.3i·4-s + (7.20 − 23.9i)5-s − 14.8·6-s + (−13.0 + 13.0i)7-s + (149. + 149. i)8-s − 77.2i·9-s + (91.1 + 169. i)10-s + 81.6·11-s + (59.2 − 59.2i)12-s + (−66.7 − 66.7i)13-s − 142. i·14-s + (42.5 − 22.8i)15-s − 929.·16-s + (347. − 347. i)17-s + ⋯ |
L(s) = 1 | + (−1.36 + 1.36i)2-s + (0.151 + 0.151i)3-s − 2.70i·4-s + (0.288 − 0.957i)5-s − 0.413·6-s + (−0.267 + 0.267i)7-s + (2.32 + 2.32i)8-s − 0.953i·9-s + (0.911 + 1.69i)10-s + 0.674·11-s + (0.411 − 0.411i)12-s + (−0.394 − 0.394i)13-s − 0.727i·14-s + (0.189 − 0.101i)15-s − 3.63·16-s + (1.20 − 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0606i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.998 + 0.0606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.712832 - 0.0216219i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.712832 - 0.0216219i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-7.20 + 23.9i)T \) |
| 7 | \( 1 + (13.0 - 13.0i)T \) |
good | 2 | \( 1 + (5.44 - 5.44i)T - 16iT^{2} \) |
| 3 | \( 1 + (-1.36 - 1.36i)T + 81iT^{2} \) |
| 11 | \( 1 - 81.6T + 1.46e4T^{2} \) |
| 13 | \( 1 + (66.7 + 66.7i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + (-347. + 347. i)T - 8.35e4iT^{2} \) |
| 19 | \( 1 + 69.2iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (-40.2 - 40.2i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + 586. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 538.T + 9.23e5T^{2} \) |
| 37 | \( 1 + (1.08e3 - 1.08e3i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 - 80.0T + 2.82e6T^{2} \) |
| 43 | \( 1 + (1.32e3 + 1.32e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + (2.67e3 - 2.67e3i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (-1.86e3 - 1.86e3i)T + 7.89e6iT^{2} \) |
| 59 | \( 1 + 1.13e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 3.39e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-1.84e3 + 1.84e3i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 - 6.93e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (823. + 823. i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 + 86.0iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-7.10e3 - 7.10e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 - 1.21e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (7.96e3 - 7.96e3i)T - 8.85e7iT^{2} \) |
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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
−16.00509243605440264800701972457, −15.05890587634365281481082686338, −13.91900972923811715004392376207, −11.98843398778409494530802538269, −9.837097542459196234488318421279, −9.314454186180193119546226084275, −8.126109513825561540817987589709, −6.59670579371973395478164001049, −5.25403100225276206955516758634, −0.821076039813503181004616344771,
1.84491461767771514370982471101, 3.46911677310301837865191265664, 7.06428180915901076193905466893, 8.309595147520267219181872086347, 9.855514672275568220699154870951, 10.56166925910822548342054412359, 11.71828803851884540961558800786, 12.99467408739213994174957367795, 14.35643168635537314753148429220, 16.47762434602355615438468556673