L(s) = 1 | + (3.55 + 2.05i)2-s + (4.91 − 1.67i)3-s + (4.43 + 7.68i)4-s − 7.61·5-s + (20.9 + 4.14i)6-s + (4.76 + 17.8i)7-s + 3.56i·8-s + (21.3 − 16.4i)9-s + (−27.0 − 15.6i)10-s − 7.17i·11-s + (34.6 + 30.3i)12-s + (−59.6 − 34.4i)13-s + (−19.7 + 73.4i)14-s + (−37.4 + 12.7i)15-s + (28.1 − 48.7i)16-s + (11.4 − 19.8i)17-s + ⋯ |
L(s) = 1 | + (1.25 + 0.726i)2-s + (0.946 − 0.322i)3-s + (0.554 + 0.960i)4-s − 0.680·5-s + (1.42 + 0.281i)6-s + (0.257 + 0.966i)7-s + 0.157i·8-s + (0.792 − 0.610i)9-s + (−0.856 − 0.494i)10-s − 0.196i·11-s + (0.834 + 0.730i)12-s + (−1.27 − 0.734i)13-s + (−0.377 + 1.40i)14-s + (−0.644 + 0.219i)15-s + (0.439 − 0.761i)16-s + (0.163 − 0.282i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.639i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.86508 + 1.03682i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.86508 + 1.03682i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-4.91 + 1.67i)T \) |
| 7 | \( 1 + (-4.76 - 17.8i)T \) |
good | 2 | \( 1 + (-3.55 - 2.05i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 + 7.61T + 125T^{2} \) |
| 11 | \( 1 + 7.17iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (59.6 + 34.4i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-11.4 + 19.8i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (86.1 - 49.7i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 - 122. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-143. + 82.8i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (11.9 - 6.89i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-126. - 219. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-159. + 276. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-156. - 271. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (35.3 - 61.3i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-254. - 146. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-230. - 399. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-573. - 331. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (359. + 622. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 92.9iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (882. + 509. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (398. - 689. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-10.7 - 18.5i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (339. + 588. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (984. - 568. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−14.80660323029227869577171628329, −13.67490137920223993073215168574, −12.57552197472699564277154158974, −11.92659253104892014483833428257, −9.772339734346017996822651128901, −8.228382067797471330322857687108, −7.34722652795042111913897060035, −5.81218976215689620022564420357, −4.32698910553452494596992697492, −2.82953816356643285908239242932,
2.38946354187092569718126446827, 4.01492087075896617182499496490, 4.63528650021564792379544905247, 7.09419694039915948806219417551, 8.396890016027267866520966648741, 10.07404981314891087881309579508, 11.10939051700528639428489775986, 12.35658781429544625258676094182, 13.25463512122954339993358137786, 14.47571406646676386546501151646