L(s) = 1 | + (0.610 + 1.05i)3-s + (0.5 + 0.866i)5-s + 0.221·7-s + (0.753 − 1.30i)9-s + 0.778·11-s + (2.5 − 4.33i)13-s + (−0.610 + 1.05i)15-s + (−3.53 − 6.12i)17-s + (−1.33 − 4.15i)19-s + (0.135 + 0.234i)21-s + (4.03 − 6.99i)23-s + (−0.499 + 0.866i)25-s + 5.50·27-s + (−0.110 + 0.192i)29-s − 2.50·31-s + ⋯ |
L(s) = 1 | + (0.352 + 0.610i)3-s + (0.223 + 0.387i)5-s + 0.0838·7-s + (0.251 − 0.435i)9-s + 0.234·11-s + (0.693 − 1.20i)13-s + (−0.157 + 0.273i)15-s + (−0.858 − 1.48i)17-s + (−0.305 − 0.952i)19-s + (0.0295 + 0.0512i)21-s + (0.842 − 1.45i)23-s + (−0.0999 + 0.173i)25-s + 1.05·27-s + (−0.0206 + 0.0356i)29-s − 0.450·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.019192303\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.019192303\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (1.33 + 4.15i)T \) |
good | 3 | \( 1 + (-0.610 - 1.05i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 - 0.221T + 7T^{2} \) |
| 11 | \( 1 - 0.778T + 11T^{2} \) |
| 13 | \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.53 + 6.12i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-4.03 + 6.99i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.110 - 0.192i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 2.50T + 31T^{2} \) |
| 37 | \( 1 + 1.90T + 37T^{2} \) |
| 41 | \( 1 + (-3.61 - 6.26i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.64 - 6.32i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.39 - 2.41i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.19 - 3.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.39 + 2.41i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.29 + 10.8i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.28 - 9.15i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.92 + 8.52i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.03 - 12.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.792 - 1.37i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9.52T + 83T^{2} \) |
| 89 | \( 1 + (1.57 - 2.71i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.18 - 5.51i)T + (-48.5 + 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.320906456611936700878542002108, −8.861617800837242837735967291091, −7.902649260838125456677668781731, −6.84049220560093037563548274499, −6.33860259763557541747952291793, −5.05805934496510621177520231966, −4.40454429756971503093498159376, −3.22361899889921561813444484197, −2.60937356737343603659511023245, −0.803641513795898249676624971418,
1.54883857646813537268135246014, 1.96800834543383774011744135318, 3.61282171090185466277545433525, 4.35521836110666599235972381823, 5.50151509610585085238729289854, 6.36641076899067438404485163383, 7.12218241522382325695604994519, 7.978537595078889150040080391854, 8.743766624621126640636664031547, 9.240324474037313296745977415134