L(s) = 1 | + (−0.161 − 1.40i)2-s + (0.734 − 0.734i)3-s + (−1.94 + 0.452i)4-s + (−1.17 − 1.90i)5-s + (−1.14 − 0.913i)6-s + 1.71·7-s + (0.949 + 2.66i)8-s + 1.92i·9-s + (−2.48 + 1.95i)10-s + (2.82 − 2.82i)11-s + (−1.09 + 1.76i)12-s + (−2.59 + 2.59i)13-s + (−0.276 − 2.40i)14-s + (−2.25 − 0.537i)15-s + (3.59 − 1.76i)16-s + 1.89i·17-s + ⋯ |
L(s) = 1 | + (−0.113 − 0.993i)2-s + (0.423 − 0.423i)3-s + (−0.974 + 0.226i)4-s + (−0.524 − 0.851i)5-s + (−0.469 − 0.372i)6-s + 0.648·7-s + (0.335 + 0.941i)8-s + 0.640i·9-s + (−0.786 + 0.617i)10-s + (0.852 − 0.852i)11-s + (−0.317 + 0.508i)12-s + (−0.719 + 0.719i)13-s + (−0.0738 − 0.643i)14-s + (−0.583 − 0.138i)15-s + (0.897 − 0.440i)16-s + 0.460i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.593411 - 0.703776i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.593411 - 0.703776i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.161 + 1.40i)T \) |
| 5 | \( 1 + (1.17 + 1.90i)T \) |
good | 3 | \( 1 + (-0.734 + 0.734i)T - 3iT^{2} \) |
| 7 | \( 1 - 1.71T + 7T^{2} \) |
| 11 | \( 1 + (-2.82 + 2.82i)T - 11iT^{2} \) |
| 13 | \( 1 + (2.59 - 2.59i)T - 13iT^{2} \) |
| 17 | \( 1 - 1.89iT - 17T^{2} \) |
| 19 | \( 1 + (-2.89 - 2.89i)T + 19iT^{2} \) |
| 23 | \( 1 - 2.00T + 23T^{2} \) |
| 29 | \( 1 + (6.72 + 6.72i)T + 29iT^{2} \) |
| 31 | \( 1 + 7.11T + 31T^{2} \) |
| 37 | \( 1 + (-2.25 - 2.25i)T + 37iT^{2} \) |
| 41 | \( 1 - 1.59iT - 41T^{2} \) |
| 43 | \( 1 + (8.06 + 8.06i)T + 43iT^{2} \) |
| 47 | \( 1 - 4.43iT - 47T^{2} \) |
| 53 | \( 1 + (-0.481 - 0.481i)T + 53iT^{2} \) |
| 59 | \( 1 + (-3.08 + 3.08i)T - 59iT^{2} \) |
| 61 | \( 1 + (-3.46 - 3.46i)T + 61iT^{2} \) |
| 67 | \( 1 + (-1.80 + 1.80i)T - 67iT^{2} \) |
| 71 | \( 1 - 0.379iT - 71T^{2} \) |
| 73 | \( 1 - 8.37T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + (8.24 - 8.24i)T - 83iT^{2} \) |
| 89 | \( 1 + 11.9iT - 89T^{2} \) |
| 97 | \( 1 - 6.50iT - 97T^{2} \) |
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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
−13.85683644678238828575415444092, −12.93114899151371344449372519681, −11.82956232471594561059082557174, −11.15559107902226723162603906496, −9.513756907126071962480608850244, −8.514573610283005270587140680518, −7.66126779756824784354080325840, −5.20701664578464931718198818531, −3.84069826645771333305387753811, −1.71650780813667433124270790332,
3.57954630959318027825433039589, 5.01099989333800356264361893145, 6.82842522520423858023852797214, 7.60831575936211312757661852472, 9.035842943502200819094537211908, 9.914829144796099010591664576025, 11.36547128909570204310854211682, 12.69351780976896352691880962471, 14.31051691936246715757610062454, 14.81509773716136642927017166041