L(s) = 1 | − 0.477·3-s − i·5-s + (2.09 + 1.61i)7-s − 2.77·9-s + 4.66i·11-s + 3.77i·13-s + 0.477i·15-s − 3.77i·17-s + 7.42·19-s + (−1 − 0.772i)21-s + 3.23i·23-s − 25-s + 2.75·27-s + 3.77·29-s − 0.954·31-s + ⋯ |
L(s) = 1 | − 0.275·3-s − 0.447i·5-s + (0.791 + 0.611i)7-s − 0.924·9-s + 1.40i·11-s + 1.04i·13-s + 0.123i·15-s − 0.914i·17-s + 1.70·19-s + (−0.218 − 0.168i)21-s + 0.674i·23-s − 0.200·25-s + 0.530·27-s + 0.700·29-s − 0.171·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.16392 + 0.571862i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.16392 + 0.571862i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (-2.09 - 1.61i)T \) |
good | 3 | \( 1 + 0.477T + 3T^{2} \) |
| 11 | \( 1 - 4.66iT - 11T^{2} \) |
| 13 | \( 1 - 3.77iT - 13T^{2} \) |
| 17 | \( 1 + 3.77iT - 17T^{2} \) |
| 19 | \( 1 - 7.42T + 19T^{2} \) |
| 23 | \( 1 - 3.23iT - 23T^{2} \) |
| 29 | \( 1 - 3.77T + 29T^{2} \) |
| 31 | \( 1 + 0.954T + 31T^{2} \) |
| 37 | \( 1 - 5.54T + 37T^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 - 12.5iT - 43T^{2} \) |
| 47 | \( 1 + 7.89T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 9.33T + 59T^{2} \) |
| 61 | \( 1 + 13.5iT - 61T^{2} \) |
| 67 | \( 1 - 6.09iT - 67T^{2} \) |
| 71 | \( 1 + 9.33iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 11.1iT - 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 - 8.22iT - 97T^{2} \) |
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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
−11.33885472474826472095003215802, −9.650117178220497669731553349803, −9.385159044612808875434642182489, −8.190408692981860181691537504483, −7.41622884967621048668975886240, −6.25883441270526384434335440090, −5.09046471307389768167625799606, −4.66652604355479149200925783203, −2.90549052726397714113491311142, −1.56437852005796828201709667723,
0.842786676772810231548423069779, 2.81400526152161706887455515732, 3.76848668091298248677010739802, 5.30666804382735750430822261582, 5.84968910094436017088564302166, 7.05443675223505984470894936379, 8.141848237525797079723059426499, 8.560248742391741641659208934817, 10.02489629967934995089770844919, 10.80241165751894725000357515047