| L(s) = 1 | + (−1.55 − 1.25i)2-s + (−2.99 + 0.172i)3-s + (0.847 + 3.90i)4-s + (−2.73 + 4.18i)5-s + (4.87 + 3.49i)6-s + (−1.87 + 1.87i)7-s + (3.58 − 7.14i)8-s + (8.94 − 1.03i)9-s + (9.51 − 3.07i)10-s + (−5.27 + 16.2i)11-s + (−3.21 − 11.5i)12-s + (10.1 + 5.17i)13-s + (5.28 − 0.566i)14-s + (7.47 − 13.0i)15-s + (−14.5 + 6.62i)16-s + (3.58 − 22.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.778 − 0.627i)2-s + (−0.998 + 0.0574i)3-s + (0.211 + 0.977i)4-s + (−0.547 + 0.836i)5-s + (0.813 + 0.582i)6-s + (−0.268 + 0.268i)7-s + (0.448 − 0.893i)8-s + (0.993 − 0.114i)9-s + (0.951 − 0.307i)10-s + (−0.479 + 1.47i)11-s + (−0.267 − 0.963i)12-s + (0.780 + 0.397i)13-s + (0.377 − 0.0404i)14-s + (0.498 − 0.867i)15-s + (−0.910 + 0.413i)16-s + (0.210 − 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00401180 - 0.0259777i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00401180 - 0.0259777i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.55 + 1.25i)T \) |
| 3 | \( 1 + (2.99 - 0.172i)T \) |
| 5 | \( 1 + (2.73 - 4.18i)T \) |
| good | 7 | \( 1 + (1.87 - 1.87i)T - 49iT^{2} \) |
| 11 | \( 1 + (5.27 - 16.2i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (-10.1 - 5.17i)T + (99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (-3.58 + 22.6i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (14.3 - 10.4i)T + (111. - 343. i)T^{2} \) |
| 23 | \( 1 + (2.89 + 5.68i)T + (-310. + 427. i)T^{2} \) |
| 29 | \( 1 + (-7.12 - 5.17i)T + (259. + 799. i)T^{2} \) |
| 31 | \( 1 + (20.3 + 28.0i)T + (-296. + 913. i)T^{2} \) |
| 37 | \( 1 + (24.2 - 47.5i)T + (-804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (35.6 - 11.5i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (38.4 + 38.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-1.85 - 11.7i)T + (-2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (14.2 + 90.0i)T + (-2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-108. + 35.1i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-21.7 + 67.0i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-6.45 + 40.7i)T + (-4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (19.3 + 14.0i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (50.2 + 98.5i)T + (-3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-17.8 - 12.9i)T + (1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (4.17 - 26.3i)T + (-6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (19.9 - 61.5i)T + (-6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-9.57 - 60.4i)T + (-8.94e3 + 2.90e3i)T^{2} \) |
| 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
−11.82347729968375052413945560322, −11.11137427086160117557728738614, −10.19767107757595670139198146960, −9.656340414870147814070567639303, −8.196597770144070119065521890042, −7.13095770679799957575861300235, −6.53897379453674709051985156996, −4.80487793641596302407547413731, −3.59592147157806805771863606255, −2.03513060484727165628227408059,
0.02116373787422723823605439197, 1.21112871112732373076889502341, 3.90486526429541228557398576758, 5.35413339882779701621468904722, 5.97884997730383361896374255710, 7.09324091129032253350246804882, 8.287066150071966655133278044279, 8.772537559438458247264180107719, 10.28149804285696300382585005536, 10.82842711638163923939931903668
