| L(s) = 1 | + (−0.446 − 1.34i)2-s + (1.72 − 0.175i)3-s + (−1.60 + 1.19i)4-s + (−2.16 + 0.575i)5-s + (−1.00 − 2.23i)6-s − 4.70i·7-s + (2.32 + 1.61i)8-s + (2.93 − 0.603i)9-s + (1.73 + 2.64i)10-s + (0.0957 − 0.294i)11-s + (−2.54 + 2.34i)12-s + (−1.61 − 4.96i)13-s + (−6.31 + 2.10i)14-s + (−3.62 + 1.37i)15-s + (1.12 − 3.83i)16-s + (−2.69 − 3.70i)17-s + ⋯ |
| L(s) = 1 | + (−0.316 − 0.948i)2-s + (0.994 − 0.101i)3-s + (−0.800 + 0.599i)4-s + (−0.966 + 0.257i)5-s + (−0.410 − 0.911i)6-s − 1.77i·7-s + (0.821 + 0.569i)8-s + (0.979 − 0.201i)9-s + (0.549 + 0.835i)10-s + (0.0288 − 0.0888i)11-s + (−0.735 + 0.677i)12-s + (−0.447 − 1.37i)13-s + (−1.68 + 0.562i)14-s + (−0.935 + 0.353i)15-s + (0.280 − 0.959i)16-s + (−0.652 − 0.898i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.630 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.491532 - 1.03282i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.491532 - 1.03282i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.446 + 1.34i)T \) |
| 3 | \( 1 + (-1.72 + 0.175i)T \) |
| 5 | \( 1 + (2.16 - 0.575i)T \) |
| good | 7 | \( 1 + 4.70iT - 7T^{2} \) |
| 11 | \( 1 + (-0.0957 + 0.294i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (1.61 + 4.96i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (2.69 + 3.70i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.88 - 3.96i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.222 - 0.685i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (2.34 - 3.22i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.15 - 2.96i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.08 - 3.33i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-6.26 + 2.03i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 2.91iT - 43T^{2} \) |
| 47 | \( 1 + (-3.23 - 2.34i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.65 + 2.27i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.546 - 1.68i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (1.48 - 4.58i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-6.49 - 8.93i)T + (-20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-0.155 - 0.112i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.448 + 1.37i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-6.71 + 9.24i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (3.47 - 2.52i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (4.46 + 1.45i)T + (72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-2.12 - 1.54i)T + (29.9 + 92.2i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.22354777251255499012354657892, −10.41023311264644972718360659306, −9.782725356678378726281636891992, −8.498735746437582539612454276623, −7.59593461274967317807641295222, −7.25990032753822983629741062133, −4.67546633049729044792887075375, −3.73076583259114455039316643129, −2.95690335595123037733671021663, −0.902931513946712535928834889299,
2.24462519405955405503425075681, 4.02561014541920868170067426631, 4.98441808772326980016911670424, 6.39727901856042576045944904269, 7.46470572834327154998951560488, 8.375476588598671460357888342188, 9.057561364163566823823605476081, 9.523607352325275517944270642914, 11.17278073085229349347586085137, 12.21754006259271653611400362092
