L(s) = 1 | + (0.309 + 0.951i)2-s + (−1.11 + 0.614i)3-s + (−0.809 + 0.587i)4-s + (−0.929 − 0.872i)6-s + (−0.809 − 0.587i)8-s + (0.335 − 0.527i)9-s + (0.844 − 1.79i)11-s + (0.542 − 1.15i)12-s + (0.309 − 0.951i)16-s + (1.45 − 1.36i)17-s + (0.605 + 0.155i)18-s + (−0.574 − 0.227i)19-s + (1.96 + 0.248i)22-s + (1.26 + 0.159i)24-s + (0.309 + 0.951i)25-s + ⋯ |
L(s) = 1 | + (0.309 + 0.951i)2-s + (−1.11 + 0.614i)3-s + (−0.809 + 0.587i)4-s + (−0.929 − 0.872i)6-s + (−0.809 − 0.587i)8-s + (0.335 − 0.527i)9-s + (0.844 − 1.79i)11-s + (0.542 − 1.15i)12-s + (0.309 − 0.951i)16-s + (1.45 − 1.36i)17-s + (0.605 + 0.155i)18-s + (−0.574 − 0.227i)19-s + (1.96 + 0.248i)22-s + (1.26 + 0.159i)24-s + (0.309 + 0.951i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8012952184\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8012952184\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 251 | \( 1 + (-0.0627 - 0.998i)T \) |
good | 3 | \( 1 + (1.11 - 0.614i)T + (0.535 - 0.844i)T^{2} \) |
| 5 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (0.992 + 0.125i)T^{2} \) |
| 11 | \( 1 + (-0.844 + 1.79i)T + (-0.637 - 0.770i)T^{2} \) |
| 13 | \( 1 + (0.425 + 0.904i)T^{2} \) |
| 17 | \( 1 + (-1.45 + 1.36i)T + (0.0627 - 0.998i)T^{2} \) |
| 19 | \( 1 + (0.574 + 0.227i)T + (0.728 + 0.684i)T^{2} \) |
| 23 | \( 1 + (-0.0627 + 0.998i)T^{2} \) |
| 29 | \( 1 + (-0.876 + 0.481i)T^{2} \) |
| 31 | \( 1 + (0.637 + 0.770i)T^{2} \) |
| 37 | \( 1 + (0.425 + 0.904i)T^{2} \) |
| 41 | \( 1 + (0.124 - 0.0157i)T + (0.968 - 0.248i)T^{2} \) |
| 43 | \( 1 + (0.929 + 1.12i)T + (-0.187 + 0.982i)T^{2} \) |
| 47 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.187 + 0.982i)T^{2} \) |
| 59 | \( 1 + (0.620 + 0.582i)T + (0.0627 + 0.998i)T^{2} \) |
| 61 | \( 1 + (0.187 + 0.982i)T^{2} \) |
| 67 | \( 1 + (-1.84 + 0.233i)T + (0.968 - 0.248i)T^{2} \) |
| 71 | \( 1 + (0.425 + 0.904i)T^{2} \) |
| 73 | \( 1 + (-0.541 - 0.297i)T + (0.535 + 0.844i)T^{2} \) |
| 79 | \( 1 + (-0.876 - 0.481i)T^{2} \) |
| 83 | \( 1 + (0.328 - 1.72i)T + (-0.929 - 0.368i)T^{2} \) |
| 89 | \( 1 + (-1.41 + 1.32i)T + (0.0627 - 0.998i)T^{2} \) |
| 97 | \( 1 + (0.0235 - 0.123i)T + (-0.929 - 0.368i)T^{2} \) |
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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
−9.360754808520031756573780138087, −8.615175979079371895043589388858, −7.81750671167175299791173222956, −6.78730055171444746318124966873, −6.18960289759680990998809706124, −5.36031250895516631435375596157, −5.04038624538058854533161908462, −3.81328905618493481575623810250, −3.17186764926558004538949443982, −0.71484707269729334135984375663,
1.25954878001978811915229432244, 1.98569930727970895796731720164, 3.46510803943351517333332388891, 4.39877521942929973163124090671, 5.10837487279901156511619513572, 6.13256383748206983812574484326, 6.53475858438595011968736086031, 7.63943526032982630940466411942, 8.542856740174160272366758279506, 9.636781980409802850790558051753