L(s) = 1 | + (1.28 + 2.21i)3-s + (0.5 − 0.866i)5-s + (−1.78 + 3.08i)9-s + (1.28 + 2.21i)11-s + 5.68·13-s + 2.56·15-s + (1.71 + 2.97i)17-s + (0.561 − 0.972i)19-s + (2.56 − 4.43i)23-s + (−0.499 − 0.866i)25-s − 1.43·27-s + 4.56·29-s + (−5.12 − 8.87i)31-s + (−3.28 + 5.68i)33-s + (−4.12 + 7.14i)37-s + ⋯ |
L(s) = 1 | + (0.739 + 1.28i)3-s + (0.223 − 0.387i)5-s + (−0.593 + 1.02i)9-s + (0.386 + 0.668i)11-s + 1.57·13-s + 0.661·15-s + (0.416 + 0.722i)17-s + (0.128 − 0.223i)19-s + (0.534 − 0.925i)23-s + (−0.0999 − 0.173i)25-s − 0.276·27-s + 0.847·29-s + (−0.920 − 1.59i)31-s + (−0.571 + 0.989i)33-s + (−0.677 + 1.17i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.671598237\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.671598237\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-1.28 - 2.21i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-1.28 - 2.21i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.68T + 13T^{2} \) |
| 17 | \( 1 + (-1.71 - 2.97i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.561 + 0.972i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.56 + 4.43i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4.56T + 29T^{2} \) |
| 31 | \( 1 + (5.12 + 8.87i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.12 - 7.14i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 7.12T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 + (-3.28 + 5.68i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.43 - 4.22i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.56 - 13.0i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.12 - 12.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-6.12 - 10.6i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.84 + 10.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (1.56 - 2.70i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−9.275006084432700848919355967668, −8.642262541205430372581140792998, −8.240214387592173005050090330598, −6.96978054196769540211173168946, −6.05294578203701796113130799455, −5.13472693775090626074507189276, −4.21266694781622261577274645182, −3.74362940032869252125142011956, −2.67882262194379045017227871740, −1.35491559699654480310465303788,
1.05184547559119673258041889096, 1.83930142815929609664240333550, 3.16831753622017040497641810319, 3.52421233915913398641555951370, 5.18165680818807270454175604096, 6.09495180027569976670453402781, 6.77329055118000203873575899924, 7.40424360090084722162621300087, 8.264864696267023615701986004034, 8.810702205361890568373761915447