L(s) = 1 | + (−0.548 + 0.950i)2-s + (0.189 − 0.328i)3-s + (0.397 + 0.689i)4-s + (−0.5 + 0.866i)5-s + (0.208 + 0.360i)6-s + 1.89·7-s − 3.06·8-s + (1.42 + 2.47i)9-s + (−0.548 − 0.950i)10-s + 0.134·11-s + 0.301·12-s + (−1.75 − 3.04i)13-s + (−1.03 + 1.79i)14-s + (0.189 + 0.328i)15-s + (0.887 − 1.53i)16-s + (0.830 − 1.43i)17-s + ⋯ |
L(s) = 1 | + (−0.388 + 0.672i)2-s + (0.109 − 0.189i)3-s + (0.198 + 0.344i)4-s + (−0.223 + 0.387i)5-s + (0.0849 + 0.147i)6-s + 0.715·7-s − 1.08·8-s + (0.476 + 0.824i)9-s + (−0.173 − 0.300i)10-s + 0.0405·11-s + 0.0871·12-s + (−0.487 − 0.843i)13-s + (−0.277 + 0.480i)14-s + (0.0489 + 0.0848i)15-s + (0.221 − 0.384i)16-s + (0.201 − 0.348i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.728251 + 0.541882i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.728251 + 0.541882i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-2.10 + 3.81i)T \) |
good | 2 | \( 1 + (0.548 - 0.950i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.189 + 0.328i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 1.89T + 7T^{2} \) |
| 11 | \( 1 - 0.134T + 11T^{2} \) |
| 13 | \( 1 + (1.75 + 3.04i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.830 + 1.43i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (2.68 + 4.65i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.48 + 4.30i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.56T + 31T^{2} \) |
| 37 | \( 1 + 1.69T + 37T^{2} \) |
| 41 | \( 1 + (5.31 - 9.20i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.25 - 7.36i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.55 - 9.62i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.132 - 0.229i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.44 + 5.97i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.58 + 7.94i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.47 - 2.55i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.664 - 1.15i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.17 + 5.49i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.733 + 1.27i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7.44T + 83T^{2} \) |
| 89 | \( 1 + (4.86 + 8.43i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.73 - 15.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−14.42752526400959679074800231018, −13.21876737934056632358172515049, −12.05151892115225309869408625381, −11.06007793221899207841359893602, −9.765192668906679572791900823604, −8.145776399454208932025343072373, −7.71926524087744570592638873788, −6.51440827645734283388967561323, −4.83039531908347741928552941327, −2.74659418026308953269554871505,
1.63166237457043358887318819985, 3.78827791908113804786578006591, 5.45370201816457705828551160740, 7.01295277276713635816245937289, 8.564824933073928753998208465424, 9.585271670426735615358706916564, 10.44773697283084872156620532584, 11.85728655962209504217303357795, 12.12111327957290101059866866900, 13.88054285000329392931953723554