L(s) = 1 | + (0.866 − 0.5i)2-s + (0.222 + 0.385i)3-s + (0.499 − 0.866i)4-s + (0.385 + 0.222i)6-s − 0.999i·8-s + (0.400 − 0.694i)9-s + (−1.07 + 0.623i)11-s + 0.445·12-s + (−0.5 − 0.866i)16-s + (0.623 − 1.07i)17-s − 0.801i·18-s + (1.56 + 0.900i)19-s + (−0.623 + 1.07i)22-s + (0.385 − 0.222i)24-s − 25-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.222 + 0.385i)3-s + (0.499 − 0.866i)4-s + (0.385 + 0.222i)6-s − 0.999i·8-s + (0.400 − 0.694i)9-s + (−1.07 + 0.623i)11-s + 0.445·12-s + (−0.5 − 0.866i)16-s + (0.623 − 1.07i)17-s − 0.801i·18-s + (1.56 + 0.900i)19-s + (−0.623 + 1.07i)22-s + (0.385 − 0.222i)24-s − 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.885434527\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.885434527\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (1.07 - 0.623i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-1.56 - 0.900i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (1.56 - 0.900i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.385 + 0.222i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.385 - 0.222i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 - 0.445iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.80iT - T^{2} \) |
| 89 | \( 1 + (-0.385 + 0.222i)T + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (1.07 + 0.623i)T + (0.5 + 0.866i)T^{2} \) |
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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
−9.947573704471808424804716838037, −9.315300174221624069830865506490, −7.87171851332043315986113897498, −7.24531904435347906534259870808, −6.17483118287280791308318139474, −5.25802507390676725488262451037, −4.61502453000986182777598518409, −3.49037766199102725360602020451, −2.86913351189514961967768555911, −1.43391724233009600052326448613,
1.90080608051119407399955955325, 2.98704715915442080982626164701, 3.86019661437208942056310414750, 5.20008772859537051104516041876, 5.46925186723726641914916442036, 6.69152068034646531773159315800, 7.47169952248103546476249707989, 8.023027117622335859611853840620, 8.730941550457539315065709006956, 10.07555760725864644340313850001