L(s) = 1 | + (0.707 − 0.707i)2-s + (0.830 − 1.52i)3-s − 1.00i·4-s + (2.23 − 0.129i)5-s + (−0.487 − 1.66i)6-s + (−2.47 − 2.47i)7-s + (−0.707 − 0.707i)8-s + (−1.62 − 2.52i)9-s + (1.48 − 1.66i)10-s + 0.319i·11-s + (−1.52 − 0.830i)12-s + (−3.95 + 3.95i)13-s − 3.49·14-s + (1.65 − 3.50i)15-s − 1.00·16-s + (4.30 − 4.30i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.479 − 0.877i)3-s − 0.500i·4-s + (0.998 − 0.0577i)5-s + (−0.199 − 0.678i)6-s + (−0.935 − 0.935i)7-s + (−0.250 − 0.250i)8-s + (−0.540 − 0.841i)9-s + (0.470 − 0.528i)10-s + 0.0963i·11-s + (−0.438 − 0.239i)12-s + (−1.09 + 1.09i)13-s − 0.935·14-s + (0.427 − 0.903i)15-s − 0.250·16-s + (1.04 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 + 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.960440 - 1.99663i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.960440 - 1.99663i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.830 + 1.52i)T \) |
| 5 | \( 1 + (-2.23 + 0.129i)T \) |
| 19 | \( 1 - iT \) |
good | 7 | \( 1 + (2.47 + 2.47i)T + 7iT^{2} \) |
| 11 | \( 1 - 0.319iT - 11T^{2} \) |
| 13 | \( 1 + (3.95 - 3.95i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.30 + 4.30i)T - 17iT^{2} \) |
| 23 | \( 1 + (-2.89 - 2.89i)T + 23iT^{2} \) |
| 29 | \( 1 - 8.60T + 29T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 37 | \( 1 + (-1.42 - 1.42i)T + 37iT^{2} \) |
| 41 | \( 1 - 7.07iT - 41T^{2} \) |
| 43 | \( 1 + (-6.45 + 6.45i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.12 + 4.12i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.15 + 1.15i)T + 53iT^{2} \) |
| 59 | \( 1 + 7.51T + 59T^{2} \) |
| 61 | \( 1 - 8.09T + 61T^{2} \) |
| 67 | \( 1 + (11.1 + 11.1i)T + 67iT^{2} \) |
| 71 | \( 1 - 13.8iT - 71T^{2} \) |
| 73 | \( 1 + (9.22 - 9.22i)T - 73iT^{2} \) |
| 79 | \( 1 + 3.00iT - 79T^{2} \) |
| 83 | \( 1 + (2.31 + 2.31i)T + 83iT^{2} \) |
| 89 | \( 1 - 4.95T + 89T^{2} \) |
| 97 | \( 1 + (2.21 + 2.21i)T + 97iT^{2} \) |
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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
−10.18077060647308623476132834042, −9.711536189448212389273075262183, −8.954447695192883974224953443400, −7.36769957285379066080094411700, −6.87965760716323799402025087340, −5.91624086159108162318278624100, −4.72588621986845550003016279070, −3.32451606678098557137736093251, −2.42785326076087705220191903074, −1.07232395508182878054290311245,
2.59650850232582216832570530297, 3.14217673276019446225749687565, 4.66581585587042042342008068404, 5.61212790734844586706733378105, 6.12031366733232210011464944448, 7.49071084699070191393376992185, 8.570146863827419312359383976574, 9.289352958125138087124082836138, 10.09372362600715795696218013473, 10.69242521181350086886426990932