L(s) = 1 | + (1.60 − 0.430i)2-s + (1.08 − 1.35i)3-s + (0.661 − 0.382i)4-s + (1.15 − 2.63i)6-s + (−0.465 − 1.73i)7-s + (−1.45 + 1.45i)8-s + (−0.661 − 2.92i)9-s + (3.12 + 1.80i)11-s + (0.198 − 1.30i)12-s + (−0.342 + 1.27i)13-s + (−1.49 − 2.59i)14-s + (−2.47 + 4.28i)16-s + (0.277 + 0.277i)17-s + (−2.32 − 4.41i)18-s + 6.25i·19-s + ⋯ |
L(s) = 1 | + (1.13 − 0.304i)2-s + (0.624 − 0.781i)3-s + (0.330 − 0.191i)4-s + (0.471 − 1.07i)6-s + (−0.175 − 0.656i)7-s + (−0.513 + 0.513i)8-s + (−0.220 − 0.975i)9-s + (0.942 + 0.544i)11-s + (0.0573 − 0.377i)12-s + (−0.0950 + 0.354i)13-s + (−0.399 − 0.692i)14-s + (−0.618 + 1.07i)16-s + (0.0671 + 0.0671i)17-s + (−0.547 − 1.04i)18-s + 1.43i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.03061 - 1.06914i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.03061 - 1.06914i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.08 + 1.35i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-1.60 + 0.430i)T + (1.73 - i)T^{2} \) |
| 7 | \( 1 + (0.465 + 1.73i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-3.12 - 1.80i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.342 - 1.27i)T + (-11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (-0.277 - 0.277i)T + 17iT^{2} \) |
| 19 | \( 1 - 6.25iT - 19T^{2} \) |
| 23 | \( 1 + (2.16 + 0.579i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-1.56 + 2.71i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.42 + 4.20i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.55 - 5.55i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1.29 + 0.744i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.10 - 1.10i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (-3.82 + 1.02i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-7.48 + 7.48i)T - 53iT^{2} \) |
| 59 | \( 1 + (0.279 + 0.483i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.96 - 5.13i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (10.8 + 2.90i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 8.01iT - 71T^{2} \) |
| 73 | \( 1 + (-1.29 - 1.29i)T + 73iT^{2} \) |
| 79 | \( 1 + (6.96 + 4.02i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.150 - 0.560i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 + (-1.37 - 5.14i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−12.10941729814631836935164303597, −11.84101187146025711660980316555, −10.21783719566277684684332981895, −9.095627859699502585609575689380, −8.019743908235567682314112967779, −6.87202145797141210086146084403, −5.91240187913987218925282310765, −4.26402657859701556839653472452, −3.48998377490408359748473657877, −1.90181723413882924293658924398,
2.82969006167080384891745720267, 3.84956480333767904856857731471, 4.96857292238856122986138619989, 5.88240479221608503386603413874, 7.14035037488592814659373013408, 8.803679109713754653709094897272, 9.200182343168605268804869180227, 10.48041154819459374579009196084, 11.64968826736465444811850735698, 12.59153205551665025320276603578