| L(s) = 1 | + (0.250 − 1.98i)2-s + (1.13 − 2.77i)3-s + (−3.87 − 0.995i)4-s + (0.801 + 4.93i)5-s + (−5.22 − 2.95i)6-s + (−6.67 + 6.67i)7-s + (−2.94 + 7.43i)8-s + (−6.41 − 6.31i)9-s + (9.99 − 0.352i)10-s + (−5.11 + 15.7i)11-s + (−7.17 + 9.62i)12-s + (8.85 + 4.51i)13-s + (11.5 + 14.9i)14-s + (14.6 + 3.38i)15-s + (14.0 + 7.71i)16-s + (1.11 − 7.06i)17-s + ⋯ |
| L(s) = 1 | + (0.125 − 0.992i)2-s + (0.379 − 0.925i)3-s + (−0.968 − 0.248i)4-s + (0.160 + 0.987i)5-s + (−0.870 − 0.492i)6-s + (−0.953 + 0.953i)7-s + (−0.368 + 0.929i)8-s + (−0.712 − 0.701i)9-s + (0.999 − 0.0352i)10-s + (−0.465 + 1.43i)11-s + (−0.597 + 0.801i)12-s + (0.680 + 0.346i)13-s + (0.826 + 1.06i)14-s + (0.974 + 0.225i)15-s + (0.876 + 0.482i)16-s + (0.0658 − 0.415i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.696380 + 0.325905i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.696380 + 0.325905i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.250 + 1.98i)T \) |
| 3 | \( 1 + (-1.13 + 2.77i)T \) |
| 5 | \( 1 + (-0.801 - 4.93i)T \) |
| good | 7 | \( 1 + (6.67 - 6.67i)T - 49iT^{2} \) |
| 11 | \( 1 + (5.11 - 15.7i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (-8.85 - 4.51i)T + (99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (-1.11 + 7.06i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (24.1 - 17.5i)T + (111. - 343. i)T^{2} \) |
| 23 | \( 1 + (19.6 + 38.6i)T + (-310. + 427. i)T^{2} \) |
| 29 | \( 1 + (12.7 + 9.22i)T + (259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-7.41 - 10.1i)T + (-296. + 913. i)T^{2} \) |
| 37 | \( 1 + (1.04 - 2.05i)T + (-804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-27.5 + 8.95i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-27.8 - 27.8i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (4.51 + 28.5i)T + (-2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (-13.0 - 82.2i)T + (-2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (17.4 - 5.67i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-18.0 + 55.4i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (5.85 - 36.9i)T + (-4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (11.3 + 8.22i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-58.8 - 115. i)T + (-3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (24.1 + 17.5i)T + (1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-2.81 + 17.7i)T + (-6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-9.89 + 30.4i)T + (-6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-6.39 - 40.3i)T + (-8.94e3 + 2.90e3i)T^{2} \) |
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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
−11.94619081669746305518468682553, −10.70742982383141275670738074370, −9.884813358844249510465849563123, −8.993183333931377714169464310912, −7.943453836374568085806247971791, −6.60241655024898316755219035760, −5.90086785856434590465799132077, −4.04675472340556446859300633762, −2.67514759047437845836280161640, −2.09902838218886378996048459983,
0.32716570918882784164646427204, 3.44211038172956632572783084873, 4.15899672333953839436786803000, 5.45883641178595979077194849921, 6.19315629364656748037692318554, 7.75958858621626809155925962461, 8.547475856151750611235511178238, 9.281452927286116535462031526764, 10.17975427032051894852603528432, 11.15008573187852427092680394463
