| L(s) = 1 | + 9.62i·2-s − 43.0·3-s + 35.3·4-s − 487.·5-s − 414. i·6-s − 870. i·7-s + 1.57e3i·8-s − 331.·9-s − 4.68e3i·10-s − 925. i·11-s − 1.52e3·12-s + 4.43e3·13-s + 8.38e3·14-s + 2.09e4·15-s − 1.06e4·16-s + 693. i·17-s + ⋯ |
| L(s) = 1 | + 0.850i·2-s − 0.920·3-s + 0.275·4-s − 1.74·5-s − 0.783i·6-s − 0.959i·7-s + 1.08i·8-s − 0.151·9-s − 1.48i·10-s − 0.209i·11-s − 0.254·12-s + 0.559·13-s + 0.816·14-s + 1.60·15-s − 0.647·16-s + 0.0342i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.835568 + 0.282935i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.835568 + 0.282935i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 61 | \( 1 + (-1.40e6 + 1.07e6i)T \) |
| good | 2 | \( 1 - 9.62iT - 128T^{2} \) |
| 3 | \( 1 + 43.0T + 2.18e3T^{2} \) |
| 5 | \( 1 + 487.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 870. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 925. iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 4.43e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 693. iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 9.51e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.10e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 1.18e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 8.33e4iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 3.63e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 2.85e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.39e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 4.55e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 5.75e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 2.92e6iT - 2.48e12T^{2} \) |
| 67 | \( 1 + 4.12e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 2.07e4iT - 9.09e12T^{2} \) |
| 73 | \( 1 + 2.27e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.54e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 6.26e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.13e7iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 1.40e7T + 8.07e13T^{2} \) |
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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
−13.95411361978043846646317378140, −12.23552379962656725377645604853, −11.32696391785607904206596596110, −10.79224748958286658649559446215, −8.426500338235783567325299243013, −7.48460172488118543759326310980, −6.54418093813842809683385682606, −5.07048344083039524175500287849, −3.58310230468842521229635031089, −0.61222550464799090507954043212,
0.73510672827596323655596642095, 2.83265509056117974234658340678, 4.24751935649060788147770918271, 5.96848549395995297933972755730, 7.39483075515842427715663683676, 8.800751711276880632519636920584, 10.58140267601304768558898747378, 11.51533094209874129233397139453, 11.88490629808681613487248032860, 12.73817988829268900641438045825
