| L(s) = 1 | + (−2.50 − 0.396i)3-s + (−3.83 − 3.21i)5-s + (1.84 − 1.84i)7-s + (−2.45 − 0.797i)9-s + (−0.224 − 0.692i)11-s + (10.9 − 5.57i)13-s + (8.31 + 9.55i)15-s + (−20.9 + 3.31i)17-s + (4.52 − 6.22i)19-s + (−5.35 + 3.88i)21-s + (−18.0 + 35.3i)23-s + (4.37 + 24.6i)25-s + (26.1 + 13.3i)27-s + (28.3 + 38.9i)29-s + (−31.6 − 22.9i)31-s + ⋯ |
| L(s) = 1 | + (−0.834 − 0.132i)3-s + (−0.766 − 0.642i)5-s + (0.263 − 0.263i)7-s + (−0.272 − 0.0885i)9-s + (−0.0204 − 0.0629i)11-s + (0.842 − 0.429i)13-s + (0.554 + 0.637i)15-s + (−1.23 + 0.194i)17-s + (0.237 − 0.327i)19-s + (−0.254 + 0.185i)21-s + (−0.783 + 1.53i)23-s + (0.174 + 0.984i)25-s + (0.968 + 0.493i)27-s + (0.976 + 1.34i)29-s + (−1.02 − 0.741i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.204007 + 0.259595i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.204007 + 0.259595i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (3.83 + 3.21i)T \) |
| good | 3 | \( 1 + (2.50 + 0.396i)T + (8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (-1.84 + 1.84i)T - 49iT^{2} \) |
| 11 | \( 1 + (0.224 + 0.692i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (-10.9 + 5.57i)T + (99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (20.9 - 3.31i)T + (274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (-4.52 + 6.22i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (18.0 - 35.3i)T + (-310. - 427. i)T^{2} \) |
| 29 | \( 1 + (-28.3 - 38.9i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (31.6 + 22.9i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-8.18 - 16.0i)T + (-804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (2.02 - 6.24i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (15.0 + 15.0i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (13.1 - 83.0i)T + (-2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (64.5 + 10.2i)T + (2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-74.4 - 24.2i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (20.0 + 61.8i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (61.2 - 9.69i)T + (4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-43.2 + 31.4i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (15.4 - 30.3i)T + (-3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-13.7 - 18.9i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-8.10 - 51.1i)T + (-6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-142. + 46.4i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-9.98 + 63.0i)T + (-8.94e3 - 2.90e3i)T^{2} \) |
| 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
−11.17626507833615206787747737532, −10.93203033826553733605137152241, −9.392892238990783372130793900958, −8.512405329265210644366772720047, −7.65039630051171562990782804910, −6.50288906216964869268283679037, −5.51467352505929585868304636261, −4.56100676238724190193567930965, −3.38238056555577952632595052728, −1.25389147002883550069842434564,
0.17594779549741674007326376364, 2.38246400070916738748448176010, 3.89373679119903196676509884113, 4.87716380760222416122023932191, 6.16575937084266951290479025276, 6.76686386887830740395518189985, 8.114023850170996282083689291304, 8.776242712723697344898601462161, 10.26931918678271245798369222204, 10.90694621092421721435495132714
