L(s) = 1 | + (0.866 − 0.5i)2-s + (0.662 + 0.382i)3-s + (0.499 − 0.866i)4-s + (−1.77 + 1.02i)5-s + 0.765·6-s − 0.999i·8-s + (−1.20 − 2.09i)9-s + (−1.02 + 1.77i)10-s + (−3.14 + 1.06i)11-s + (0.662 − 0.382i)12-s − 6.59·13-s − 1.57·15-s + (−0.5 − 0.866i)16-s + (−1.80 + 3.12i)17-s + (−2.09 − 1.20i)18-s + (−0.439 − 0.760i)19-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.382 + 0.220i)3-s + (0.249 − 0.433i)4-s + (−0.794 + 0.458i)5-s + 0.312·6-s − 0.353i·8-s + (−0.402 − 0.696i)9-s + (−0.324 + 0.561i)10-s + (−0.947 + 0.320i)11-s + (0.191 − 0.110i)12-s − 1.82·13-s − 0.405·15-s + (−0.125 − 0.216i)16-s + (−0.437 + 0.758i)17-s + (−0.492 − 0.284i)18-s + (−0.100 − 0.174i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1953826406\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1953826406\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (3.14 - 1.06i)T \) |
good | 3 | \( 1 + (-0.662 - 0.382i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1.77 - 1.02i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 6.59T + 13T^{2} \) |
| 17 | \( 1 + (1.80 - 3.12i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.439 + 0.760i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.30 + 5.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6.18iT - 29T^{2} \) |
| 31 | \( 1 + (-5.61 - 3.24i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.52 - 9.56i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.36T + 41T^{2} \) |
| 43 | \( 1 + 7.81iT - 43T^{2} \) |
| 47 | \( 1 + (4.97 - 2.87i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.214 - 0.372i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.78 - 1.60i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.66 + 9.81i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.48 - 2.56i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2.13T + 71T^{2} \) |
| 73 | \( 1 + (5.41 - 9.37i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.34 - 4.24i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 + (-5.82 + 3.36i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 9.23iT - 97T^{2} \) |
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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
−9.846660722939183826134390438986, −8.518253079738498566862800220721, −7.81784616945729540244082621223, −6.90237906144358141759177760932, −6.02909893950951084359809923454, −4.76939809660377362862687296509, −4.17933899100886923820834745669, −2.98355446919923031388627807095, −2.38506824352043387972113675950, −0.06020440305025750954643603355,
2.27603490392216699910600767283, 3.05965492268437975483436058087, 4.42033275686206931425850289425, 5.02006339931403755229229592445, 5.89059110525151114591851275062, 7.40870091645227934930751695967, 7.57530187386227234265551627355, 8.354374435994278669372668676518, 9.344961652840493783432874705335, 10.32347877954468722002589127352