L(s) = 1 | + 39.2·2-s − 12.8i·3-s + 1.02e3·4-s − 2.41e3i·5-s − 503. i·6-s + 2.30e3i·7-s + 2.02e4·8-s + 1.95e4·9-s − 9.47e4i·10-s + 8.14e4i·11-s − 1.31e4i·12-s − 5.93e4·13-s + 9.03e4i·14-s − 3.09e4·15-s + 2.68e5·16-s + (−2.89e5 − 1.87e5i)17-s + ⋯ |
L(s) = 1 | + 1.73·2-s − 0.0915i·3-s + 2.00·4-s − 1.72i·5-s − 0.158i·6-s + 0.362i·7-s + 1.74·8-s + 0.991·9-s − 2.99i·10-s + 1.67i·11-s − 0.183i·12-s − 0.576·13-s + 0.628i·14-s − 0.158·15-s + 1.02·16-s + (−0.839 − 0.543i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(4.20320 - 1.24243i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.20320 - 1.24243i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (2.89e5 + 1.87e5i)T \) |
good | 2 | \( 1 - 39.2T + 512T^{2} \) |
| 3 | \( 1 + 12.8iT - 1.96e4T^{2} \) |
| 5 | \( 1 + 2.41e3iT - 1.95e6T^{2} \) |
| 7 | \( 1 - 2.30e3iT - 4.03e7T^{2} \) |
| 11 | \( 1 - 8.14e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 + 5.93e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 3.66e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.24e6iT - 1.80e12T^{2} \) |
| 29 | \( 1 + 1.35e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 - 4.78e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 - 8.36e6iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 2.02e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 - 2.33e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.90e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.82e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 2.09e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.68e8iT - 1.16e16T^{2} \) |
| 67 | \( 1 + 1.42e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.49e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 - 2.50e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 + 1.12e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 + 5.17e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.46e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 7.57e8iT - 7.60e17T^{2} \) |
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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
−15.99125871188734003631075636464, −15.29916329700073775945807218595, −13.56203962492967231169901651906, −12.58982101654063360194529498157, −12.02764314515844533989069111186, −9.519397495315872494216591735420, −7.22026508346836077068609258340, −5.15143163604946407389243489859, −4.39439223587545790291293910052, −1.83091115279980781942673744034,
2.72624736277103472784973707427, 3.99939777622717558710339755252, 6.08972515061319450906711888591, 7.19168898322805231070951159806, 10.51276058318630736938637816579, 11.42076822807046790791682388026, 13.16343242665955402336126319310, 14.16791976926534245683124472630, 15.05558814328202019493996722717, 16.22277087366625083124567774201