L(s) = 1 | + (−0.908 − 1.08i)2-s + (−0.347 + 1.96i)4-s + 0.863·5-s − i·7-s + (2.44 − 1.41i)8-s + (−0.784 − 0.935i)10-s − 2.62i·11-s − 1.01i·13-s + (−1.08 + 0.908i)14-s + (−3.75 − 1.36i)16-s − 2.34i·17-s − 4.09·19-s + (−0.300 + 1.70i)20-s + (−2.84 + 2.39i)22-s + 6.23·23-s + ⋯ |
L(s) = 1 | + (−0.642 − 0.766i)2-s + (−0.173 + 0.984i)4-s + 0.386·5-s − 0.377i·7-s + (0.866 − 0.499i)8-s + (−0.248 − 0.295i)10-s − 0.792i·11-s − 0.282i·13-s + (−0.289 + 0.242i)14-s + (−0.939 − 0.342i)16-s − 0.568i·17-s − 0.939·19-s + (−0.0671 + 0.380i)20-s + (−0.607 + 0.509i)22-s + 1.29·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8761871952\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8761871952\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.908 + 1.08i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 - 0.863T + 5T^{2} \) |
| 11 | \( 1 + 2.62iT - 11T^{2} \) |
| 13 | \( 1 + 1.01iT - 13T^{2} \) |
| 17 | \( 1 + 2.34iT - 17T^{2} \) |
| 19 | \( 1 + 4.09T + 19T^{2} \) |
| 23 | \( 1 - 6.23T + 23T^{2} \) |
| 29 | \( 1 + 1.57T + 29T^{2} \) |
| 31 | \( 1 - 6.29iT - 31T^{2} \) |
| 37 | \( 1 + 5.18iT - 37T^{2} \) |
| 41 | \( 1 + 10.1iT - 41T^{2} \) |
| 43 | \( 1 + 0.496T + 43T^{2} \) |
| 47 | \( 1 + 5.24T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 4.40iT - 59T^{2} \) |
| 61 | \( 1 + 9.59iT - 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 + 2.11T + 71T^{2} \) |
| 73 | \( 1 + 8.27T + 73T^{2} \) |
| 79 | \( 1 + 9.44iT - 79T^{2} \) |
| 83 | \( 1 + 6.62iT - 83T^{2} \) |
| 89 | \( 1 + 10.2iT - 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−8.988685447607747849927857030105, −8.738084706647096189464463290290, −7.61024582693629142487993533297, −6.97543435366151514001460120032, −5.86511313586093762261589293181, −4.80851474421274156224725684723, −3.73893405471473611908945273566, −2.89376766993530533529403232369, −1.75607297059119590751002838976, −0.43462258558654994223710771503,
1.47009861454043742926921113971, 2.47263752979043014007339716677, 4.16051856365152977189514758684, 4.99997600739343564880219551072, 5.94883801090069220165317860510, 6.56627634627830243344746267335, 7.42086722889637938794202407306, 8.241587496372291061870595058082, 8.976271886588918619861582421304, 9.688505165008886335868045799451