| L(s) = 1 | + 6.90i·2-s + 11.0i·3-s + 16.3·4-s − 107. i·5-s − 76.4·6-s + 594. i·7-s + 554. i·8-s + 606.·9-s + 742.·10-s − 326.·11-s + 181. i·12-s − 3.32e3·13-s − 4.10e3·14-s + 1.19e3·15-s − 2.77e3·16-s + 1.73e3·17-s + ⋯ |
| L(s) = 1 | + 0.862i·2-s + 0.410i·3-s + 0.255·4-s − 0.860i·5-s − 0.354·6-s + 1.73i·7-s + 1.08i·8-s + 0.831·9-s + 0.742·10-s − 0.245·11-s + 0.104i·12-s − 1.51·13-s − 1.49·14-s + 0.352·15-s − 0.678·16-s + 0.352·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 - 0.639i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.769 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.609772 + 1.68822i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.609772 + 1.68822i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-6.11e4 - 5.08e4i)T \) |
| good | 2 | \( 1 - 6.90iT - 64T^{2} \) |
| 3 | \( 1 - 11.0iT - 729T^{2} \) |
| 5 | \( 1 + 107. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 594. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 326.T + 1.77e6T^{2} \) |
| 13 | \( 1 + 3.32e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 1.73e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 6.99e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 7.59e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + 1.00e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.58e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 3.79e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 1.83e4T + 4.75e9T^{2} \) |
| 47 | \( 1 - 1.30e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 2.72e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 2.16e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 3.44e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 2.27e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 2.81e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 2.94e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 4.97e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 1.43e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 5.23e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 4.30e5T + 8.32e11T^{2} \) |
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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
−15.37002200661033244566043649144, −14.51153727582252772451707735961, −12.55439266509926728154469170818, −11.94109368121151074937118152359, −10.00801402644735660856220822585, −8.794194196462902254947099353426, −7.59680484253913938550806498325, −5.86071756915218075604852564575, −4.88152901737494442264910615984, −2.24546500496106176198308224729,
0.855213602883476546646458016994, 2.60368732741431465554009586666, 4.20423708065706986589797669495, 7.01771737868492486808679724581, 7.26712829765033705539250479986, 10.09327134676063999862379853646, 10.41905509904692316856155381150, 11.76765492689962454337848632351, 12.95132747284415226769519076960, 13.96761902507883942650484124212
