L(s) = 1 | − 38.1i·2-s + (−95.6 + 95.6i)3-s − 944.·4-s + (−23.8 + 23.8i)5-s + (3.65e3 + 3.65e3i)6-s + (3.40e3 + 3.40e3i)7-s + 1.65e4i·8-s + 1.37e3i·9-s + (909. + 909. i)10-s + (−3.85e3 − 3.85e3i)11-s + (9.04e4 − 9.04e4i)12-s + 8.45e4·13-s + (1.29e5 − 1.29e5i)14-s − 4.55e3i·15-s + 1.46e5·16-s + (−1.50e5 + 3.09e5i)17-s + ⋯ |
L(s) = 1 | − 1.68i·2-s + (−0.681 + 0.681i)3-s − 1.84·4-s + (−0.0170 + 0.0170i)5-s + (1.15 + 1.15i)6-s + (0.536 + 0.536i)7-s + 1.42i·8-s + 0.0699i·9-s + (0.0287 + 0.0287i)10-s + (−0.0793 − 0.0793i)11-s + (1.25 − 1.25i)12-s + 0.821·13-s + (0.904 − 0.904i)14-s − 0.0232i·15-s + 0.560·16-s + (−0.438 + 0.898i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.939266 + 0.0987373i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.939266 + 0.0987373i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (1.50e5 - 3.09e5i)T \) |
good | 2 | \( 1 + 38.1iT - 512T^{2} \) |
| 3 | \( 1 + (95.6 - 95.6i)T - 1.96e4iT^{2} \) |
| 5 | \( 1 + (23.8 - 23.8i)T - 1.95e6iT^{2} \) |
| 7 | \( 1 + (-3.40e3 - 3.40e3i)T + 4.03e7iT^{2} \) |
| 11 | \( 1 + (3.85e3 + 3.85e3i)T + 2.35e9iT^{2} \) |
| 13 | \( 1 - 8.45e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 7.31e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 + (-8.57e5 - 8.57e5i)T + 1.80e12iT^{2} \) |
| 29 | \( 1 + (-3.25e6 + 3.25e6i)T - 1.45e13iT^{2} \) |
| 31 | \( 1 + (5.70e6 - 5.70e6i)T - 2.64e13iT^{2} \) |
| 37 | \( 1 + (6.44e6 - 6.44e6i)T - 1.29e14iT^{2} \) |
| 41 | \( 1 + (1.74e7 + 1.74e7i)T + 3.27e14iT^{2} \) |
| 43 | \( 1 + 1.54e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 - 3.04e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.14e5iT - 3.29e15T^{2} \) |
| 59 | \( 1 - 1.32e8iT - 8.66e15T^{2} \) |
| 61 | \( 1 + (4.51e7 + 4.51e7i)T + 1.16e16iT^{2} \) |
| 67 | \( 1 + 4.10e6T + 2.72e16T^{2} \) |
| 71 | \( 1 + (-9.21e7 + 9.21e7i)T - 4.58e16iT^{2} \) |
| 73 | \( 1 + (8.35e7 - 8.35e7i)T - 5.88e16iT^{2} \) |
| 79 | \( 1 + (-1.97e8 - 1.97e8i)T + 1.19e17iT^{2} \) |
| 83 | \( 1 - 4.27e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + 5.00e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (7.92e8 - 7.92e8i)T - 7.60e17iT^{2} \) |
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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
−17.09342244129047230179821525621, −15.46227189863564922018144065122, −13.63093030590261947690324311337, −12.18389858227757778386139009719, −11.10554913957941738064326993234, −10.30095079127147871875245988773, −8.691183338093926428678970911794, −5.36409234191138044306684704765, −3.76696916657282237257509529828, −1.67442194652201133986155890494,
0.54302074007206039399461533868, 4.79983903915487808034170975847, 6.37496495904557009213784521743, 7.33315194977624646439921215045, 8.873583818499447942952041671502, 11.22619637568744993053548767534, 13.05243766511914672257365988037, 14.20877489887270722048286799067, 15.56107892402711394771975500603, 16.72191026030510827150169444460