L(s) = 1 | + (0.879 + 1.52i)3-s + (−0.452 + 0.784i)5-s + (−0.237 − 2.63i)7-s + (−0.0471 + 0.0816i)9-s + (0.358 + 0.620i)11-s + 13-s − 1.59·15-s + (−1.17 − 2.03i)17-s + (3.31 − 5.74i)19-s + (3.80 − 2.67i)21-s + (1.87 − 3.25i)23-s + (2.08 + 3.61i)25-s + 5.11·27-s + 3.25·29-s + (0.785 + 1.36i)31-s + ⋯ |
L(s) = 1 | + (0.507 + 0.879i)3-s + (−0.202 + 0.350i)5-s + (−0.0898 − 0.995i)7-s + (−0.0157 + 0.0272i)9-s + (0.107 + 0.187i)11-s + 0.277·13-s − 0.411·15-s + (−0.285 − 0.494i)17-s + (0.761 − 1.31i)19-s + (0.830 − 0.584i)21-s + (0.391 − 0.678i)23-s + (0.417 + 0.723i)25-s + 0.983·27-s + 0.604·29-s + (0.141 + 0.244i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.025310586\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.025310586\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.237 + 2.63i)T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + (-0.879 - 1.52i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.452 - 0.784i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.358 - 0.620i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.17 + 2.03i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.31 + 5.74i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.87 + 3.25i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.25T + 29T^{2} \) |
| 31 | \( 1 + (-0.785 - 1.36i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.60 - 4.51i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4.92T + 41T^{2} \) |
| 43 | \( 1 - 9.43T + 43T^{2} \) |
| 47 | \( 1 + (4.15 - 7.20i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.04 + 12.1i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.358 - 0.620i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.82 + 10.0i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.69 - 8.13i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + (-1.73 - 3.00i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.50 + 11.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.54T + 83T^{2} \) |
| 89 | \( 1 + (6.02 - 10.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 7.43T + 97T^{2} \) |
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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
−9.501909361088916866424086955479, −8.972576143387456445877214197526, −7.964589405408792661419896231143, −7.02283858039324337824125882210, −6.56844367255273669291696568038, −5.00303644639597248520019333659, −4.43474584542590327585052638079, −3.46788370489166103324348031993, −2.79128297043353683943189203587, −0.927712841614650519625416489895,
1.21769271707525856776267933019, 2.23806385677537761461986923853, 3.22600546699323344128827790776, 4.39207735109015201215421249896, 5.54635862850692990946959183808, 6.19701085291063013359308071023, 7.23621478579930500696646571128, 7.983957467475380595769794614760, 8.572175747946711096424408620441, 9.236670978416407231405688104031