| L(s) = 1 | + (−7.92 + 1.11i)2-s + (8.95 − 8.95i)3-s + (61.5 − 17.6i)4-s + (−127. + 127. i)5-s + (−60.9 + 80.8i)6-s − 458.·7-s + (−467. + 208. i)8-s + 568. i·9-s + (865. − 1.14e3i)10-s + (912. + 912. i)11-s + (392. − 708. i)12-s + (−1.92e3 − 1.92e3i)13-s + (3.63e3 − 512. i)14-s + 2.27e3i·15-s + (3.47e3 − 2.17e3i)16-s − 1.61e3·17-s + ⋯ |
| L(s) = 1 | + (−0.990 + 0.139i)2-s + (0.331 − 0.331i)3-s + (0.961 − 0.276i)4-s + (−1.01 + 1.01i)5-s + (−0.281 + 0.374i)6-s − 1.33·7-s + (−0.913 + 0.407i)8-s + 0.780i·9-s + (0.865 − 1.14i)10-s + (0.685 + 0.685i)11-s + (0.226 − 0.410i)12-s + (−0.876 − 0.876i)13-s + (1.32 − 0.186i)14-s + 0.674i·15-s + (0.847 − 0.531i)16-s − 0.329·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 - 0.579i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.815 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.113770 + 0.356433i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.113770 + 0.356433i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (7.92 - 1.11i)T \) |
| good | 3 | \( 1 + (-8.95 + 8.95i)T - 729iT^{2} \) |
| 5 | \( 1 + (127. - 127. i)T - 1.56e4iT^{2} \) |
| 7 | \( 1 + 458.T + 1.17e5T^{2} \) |
| 11 | \( 1 + (-912. - 912. i)T + 1.77e6iT^{2} \) |
| 13 | \( 1 + (1.92e3 + 1.92e3i)T + 4.82e6iT^{2} \) |
| 17 | \( 1 + 1.61e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + (784. - 784. i)T - 4.70e7iT^{2} \) |
| 23 | \( 1 - 1.12e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (2.32e4 + 2.32e4i)T + 5.94e8iT^{2} \) |
| 31 | \( 1 - 2.45e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (-1.94e4 + 1.94e4i)T - 2.56e9iT^{2} \) |
| 41 | \( 1 - 1.03e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (4.42e4 + 4.42e4i)T + 6.32e9iT^{2} \) |
| 47 | \( 1 - 1.68e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (1.24e5 - 1.24e5i)T - 2.21e10iT^{2} \) |
| 59 | \( 1 + (8.48e4 + 8.48e4i)T + 4.21e10iT^{2} \) |
| 61 | \( 1 + (-1.10e5 - 1.10e5i)T + 5.15e10iT^{2} \) |
| 67 | \( 1 + (-2.68e5 + 2.68e5i)T - 9.04e10iT^{2} \) |
| 71 | \( 1 - 1.62e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 2.85e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 4.82e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (3.04e4 - 3.04e4i)T - 3.26e11iT^{2} \) |
| 89 | \( 1 + 4.09e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.45e6T + 8.32e11T^{2} \) |
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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
−18.64367396165595003389617473157, −17.06112432062277606480093194592, −15.70665046771436108205396264296, −14.74519187835831785549571684418, −12.61955780768232644345763021282, −11.02443822824402014758864955721, −9.676216411850673956951962348861, −7.75327480009997581799611019974, −6.78270163813573482122524860827, −2.88708472455068552461074071590,
0.33244283634086904706063069171, 3.61736216764498603811428040112, 6.83152516540770944025868210848, 8.777606515469921628887312880751, 9.515909136515368872774792636834, 11.60440409120746605881860726768, 12.67654765822137877728981508363, 15.09334121296663330968599975250, 16.24468232528715195338325696947, 16.92397488514224451686584844551
