L(s) = 1 | + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)15-s + 16-s + 17-s + (−0.5 + 0.866i)18-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)20-s + ⋯ |
L(s) = 1 | + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)15-s + 16-s + 17-s + (−0.5 + 0.866i)18-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.148015607 - 0.8188942059i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.148015607 - 0.8188942059i\) |
\(L(1)\) |
\(\approx\) |
\(1.324140942 - 0.5538359036i\) |
\(L(1)\) |
\(\approx\) |
\(1.324140942 - 0.5538359036i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−30.7352749688962183214168912400, −29.757755708472368384550495042579, −28.59154619474868728729571806893, −27.6012162402327865243824592470, −26.29002492667162924363360691024, −25.50919808371512492434275387459, −23.75904156274503170466361931295, −23.064481889964904407272401204498, −22.30903122451837835094768564156, −21.27784840915800154976349226479, −20.36258934379406551129103947414, −19.01480120652363280819690047805, −17.44363626874521811052036685939, −16.20399976267742322891809060690, −15.1343944144563409415163901230, −14.69224326813625806662058543171, −12.9980272394096008766292260042, −11.71071192622078270597637175774, −10.90025939289953996100737540494, −9.82937728956556490368271779393, −7.63344440395395449220792125942, −6.42154069036146391351012618253, −5.08826070635576404873604375864, −3.95785702087034736680176180707, −2.722101293863071919010828301635,
1.405636617404764477734188462, 3.29031156911320484484665842589, 4.98763758184693891409409141227, 5.87420098746870088055182803777, 7.33751009391216358856833881797, 8.381430361918780691701235484941, 10.68156838019510153343574421611, 11.77647998029252136527854527850, 12.65583821523367626736176633973, 13.44148487082425743935978367057, 14.7637756229022836193714376881, 16.34197866867279063176690331877, 16.77261492365786552796222764544, 18.61771021176854248056356575735, 19.59073441995526987318334539427, 20.75036921024340250118753468951, 21.81166872814354990743452588249, 23.3189513259962473894532520012, 23.5517289963570474965339744185, 24.68906067315700481723117121747, 25.420349260764780037624558447486, 27.29657764265689399405342582661, 28.55637348500254695683264622029, 29.27991385875920815603204497707, 30.228574748706561003059116558230