| L(s) = 1 | + (−1.08 − 0.901i)2-s + (2.37 − 0.417i)3-s + (0.375 + 1.96i)4-s + (0.299 − 3.42i)5-s + (−2.95 − 1.68i)6-s + (−0.746 − 0.890i)7-s + (1.36 − 2.47i)8-s + (2.62 − 0.954i)9-s + (−3.41 + 3.46i)10-s + (−2.42 − 1.39i)11-s + (1.71 + 4.49i)12-s + (−3.37 + 1.57i)13-s + (0.0119 + 1.64i)14-s + (−0.721 − 8.24i)15-s + (−3.71 + 1.47i)16-s + (6.73 + 3.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.770 − 0.637i)2-s + (1.36 − 0.241i)3-s + (0.187 + 0.982i)4-s + (0.134 − 1.53i)5-s + (−1.20 − 0.685i)6-s + (−0.282 − 0.336i)7-s + (0.481 − 0.876i)8-s + (0.874 − 0.318i)9-s + (−1.07 + 1.09i)10-s + (−0.730 − 0.421i)11-s + (0.494 + 1.29i)12-s + (−0.935 + 0.436i)13-s + (0.00318 + 0.439i)14-s + (−0.186 − 2.12i)15-s + (−0.929 + 0.369i)16-s + (1.63 + 0.761i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.273 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.785928 - 1.04035i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.785928 - 1.04035i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.08 + 0.901i)T \) |
| 37 | \( 1 + (-5.96 + 1.16i)T \) |
| good | 3 | \( 1 + (-2.37 + 0.417i)T + (2.81 - 1.02i)T^{2} \) |
| 5 | \( 1 + (-0.299 + 3.42i)T + (-4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (0.746 + 0.890i)T + (-1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (2.42 + 1.39i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.37 - 1.57i)T + (8.35 - 9.95i)T^{2} \) |
| 17 | \( 1 + (-6.73 - 3.13i)T + (10.9 + 13.0i)T^{2} \) |
| 19 | \( 1 + (-1.15 + 1.65i)T + (-6.49 - 17.8i)T^{2} \) |
| 23 | \( 1 + (-3.54 - 0.949i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-0.993 + 0.266i)T + (25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (-2.80 - 2.80i)T + 31iT^{2} \) |
| 41 | \( 1 + (-0.140 + 0.386i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (3.20 + 3.20i)T + 43iT^{2} \) |
| 47 | \( 1 + (8.81 - 5.08i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.01 + 7.16i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (0.111 + 1.27i)T + (-58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (-6.09 - 13.0i)T + (-39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (-2.24 - 2.67i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-7.67 + 1.35i)T + (66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 - 13.4iT - 73T^{2} \) |
| 79 | \( 1 + (1.18 - 13.5i)T + (-77.7 - 13.7i)T^{2} \) |
| 83 | \( 1 + (3.20 - 1.16i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-12.2 + 1.07i)T + (87.6 - 15.4i)T^{2} \) |
| 97 | \( 1 + (-4.13 + 15.4i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−11.58915751153754437309191303038, −10.06368038787553447771379581697, −9.564657891782181506393760637510, −8.565757877856989356535623307525, −8.131894096232122636576635228562, −7.18987877176212113513069571155, −5.19010011569892567627971046173, −3.78567312735015458110464754498, −2.62051947539557760753157200879, −1.17145560742349149365700435382,
2.47554756160003424394954231761, 3.12348794823766501181381710292, 5.18279187130821729468741913029, 6.49007990876482674945716641333, 7.65490450711137000778441347547, 7.86314886538568533264234615229, 9.395361155271789397319996033461, 9.882825338584864394520529738287, 10.55628373817345675074003435257, 11.86181862237155456045187837731
