L(s) = 1 | + (−2.59 + 1.5i)3-s + (−1.73 + 2i)7-s + (3 − 5.19i)9-s + (0.5 + 0.866i)11-s − 2i·13-s + (2.59 − 1.5i)17-s + (2.5 − 4.33i)19-s + (1.5 − 7.79i)21-s + (−2.59 − 1.5i)23-s + 9i·27-s + 6·29-s + (0.5 + 0.866i)31-s + (−2.59 − 1.5i)33-s + (4.33 + 2.5i)37-s + (3 + 5.19i)39-s + ⋯ |
L(s) = 1 | + (−1.49 + 0.866i)3-s + (−0.654 + 0.755i)7-s + (1 − 1.73i)9-s + (0.150 + 0.261i)11-s − 0.554i·13-s + (0.630 − 0.363i)17-s + (0.573 − 0.993i)19-s + (0.327 − 1.70i)21-s + (−0.541 − 0.312i)23-s + 1.73i·27-s + 1.11·29-s + (0.0898 + 0.155i)31-s + (−0.452 − 0.261i)33-s + (0.711 + 0.410i)37-s + (0.480 + 0.832i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8246827824\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8246827824\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.73 - 2i)T \) |
good | 3 | \( 1 + (2.59 - 1.5i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + (-2.59 + 1.5i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.59 + 1.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.33 - 2.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.79 + 4.5i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.52 + 5.5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + (6.06 - 3.5i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + (4.5 - 7.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−9.940168493706930397509956812364, −9.192704640679298050728500443626, −8.160860134984110833243419192882, −6.84444616856926999193497011846, −6.33197865983497823130620806715, −5.35007060963551862725872608743, −4.97111835481233052755104832977, −3.81222719987911031909204887147, −2.74268268111958846717805403624, −0.75232851537643708635736921052,
0.67903937486665138579953681563, 1.73299149040166484438055532257, 3.43610676909688336185443448461, 4.46947973162141114562289380629, 5.57693611473715008357153401594, 6.14879860664653064886399727095, 6.89271740989291747044891713116, 7.52043515052578033642782307792, 8.426831332028785301998755143517, 9.831946640077834808168856306949