L(s) = 1 | + (−1.06 + 1.83i)2-s + (−1.25 − 2.17i)4-s − 5-s + (−0.733 − 1.27i)7-s + 1.07·8-s + (1.06 − 1.83i)10-s + (0.172 − 0.298i)11-s + (1.76 − 3.14i)13-s + 3.11·14-s + (1.36 − 2.36i)16-s + (2.56 + 4.44i)17-s + (−0.253 − 0.438i)19-s + (1.25 + 2.17i)20-s + (0.365 + 0.632i)22-s + (0.537 − 0.930i)23-s + ⋯ |
L(s) = 1 | + (−0.750 + 1.29i)2-s + (−0.626 − 1.08i)4-s − 0.447·5-s + (−0.277 − 0.480i)7-s + 0.380·8-s + (0.335 − 0.581i)10-s + (0.0518 − 0.0898i)11-s + (0.490 − 0.871i)13-s + 0.832·14-s + (0.341 − 0.591i)16-s + (0.622 + 1.07i)17-s + (−0.0580 − 0.100i)19-s + (0.280 + 0.485i)20-s + (0.0778 + 0.134i)22-s + (0.112 − 0.194i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.749059 + 0.312500i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.749059 + 0.312500i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + (-1.76 + 3.14i)T \) |
good | 2 | \( 1 + (1.06 - 1.83i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (0.733 + 1.27i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.172 + 0.298i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.56 - 4.44i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.253 + 0.438i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.537 + 0.930i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.02 + 6.97i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 9.17T + 31T^{2} \) |
| 37 | \( 1 + (1.36 - 2.36i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.06 + 5.30i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.93 - 5.08i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 4.26T + 47T^{2} \) |
| 53 | \( 1 - 4.07T + 53T^{2} \) |
| 59 | \( 1 + (2.53 + 4.38i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.58 - 4.47i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.598 - 1.03i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.222 - 0.386i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 7.52T + 73T^{2} \) |
| 79 | \( 1 - 0.834T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 + (-5.97 + 10.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.700 - 1.21i)T + (-48.5 + 84.0i)T^{2} \) |
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
−10.36982084528965871752716593721, −9.920718376202744020713092293342, −8.610198755981969259115489825980, −8.170702306229724285185661143358, −7.37417001891157676813130459025, −6.39724126600876301270343782023, −5.74781761757980749438009697177, −4.38464944805874712355643041683, −3.12398047394544547838782229573, −0.75487914430656409040485524777,
1.10492623093241095047129026657, 2.55963128393530351015437525547, 3.45962494245081517311096089939, 4.68322111195474375075252477910, 6.09051467153307138622797421414, 7.20921483313004448559741001117, 8.375902329672482517400714404557, 9.037130095597541441943883650235, 9.767521689524044859069223751027, 10.60321896016353712209046775911