L(s) = 1 | + (1.28 − 2.21i)2-s + (−0.5 + 0.866i)3-s + (−2.28 − 3.95i)4-s − 0.561·5-s + (1.28 + 2.21i)6-s + (−1.78 − 3.08i)7-s − 6.56·8-s + (−0.499 − 0.866i)9-s + (−0.719 + 1.24i)10-s + (−1 + 1.73i)11-s + 4.56·12-s − 9.12·14-s + (0.280 − 0.486i)15-s + (−3.84 + 6.65i)16-s + (−1.28 − 2.21i)17-s − 2.56·18-s + ⋯ |
L(s) = 1 | + (0.905 − 1.56i)2-s + (−0.288 + 0.499i)3-s + (−1.14 − 1.97i)4-s − 0.251·5-s + (0.522 + 0.905i)6-s + (−0.673 − 1.16i)7-s − 2.31·8-s + (−0.166 − 0.288i)9-s + (−0.227 + 0.393i)10-s + (−0.301 + 0.522i)11-s + 1.31·12-s − 2.43·14-s + (0.0724 − 0.125i)15-s + (−0.960 + 1.66i)16-s + (−0.310 − 0.538i)17-s − 0.603·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.304754 + 1.16724i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.304754 + 1.16724i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-1.28 + 2.21i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 0.561T + 5T^{2} \) |
| 7 | \( 1 + (1.78 + 3.08i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.28 + 2.21i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.561 + 0.972i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.84 + 4.92i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 1.56T + 31T^{2} \) |
| 37 | \( 1 + (-1.71 + 2.97i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.28 + 2.21i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.219 + 0.379i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 8.24T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + (5.56 + 9.63i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.06 + 10.4i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.219 + 0.379i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-7 - 12.1i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 1.87T + 73T^{2} \) |
| 79 | \( 1 - 9.56T + 79T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 + (-6.56 + 11.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.21 + 3.84i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.51837116571101508894078048619, −9.947290174622120162720620225379, −9.260354150233821760845694221966, −7.61643638444700356476380948937, −6.37901254865646015753273483190, −5.18508400443900215780810535581, −4.23115734252547895190767245394, −3.66302251458454301614862436545, −2.38450978439024155979693654599, −0.54296060004910128312835454682,
2.74794410724370095618615346792, 4.01362408826395669446187411039, 5.25738935679637812614144233208, 6.00364067197508502832458875691, 6.54295540324409244938946421240, 7.63661169518696744267818112828, 8.405938749406118572164104092231, 9.137941449967986681161789006876, 10.63692158689389082826484948782, 12.10886074694302035525209266262