L(s) = 1 | + (0.766 + 0.642i)2-s + (1.78 − 1.49i)3-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s + 2.32·6-s + (2.45 + 0.892i)7-s + (−0.500 + 0.866i)8-s + (0.420 − 2.38i)9-s + (−0.5 − 0.866i)10-s + (0.947 − 1.64i)11-s + (1.78 + 1.49i)12-s + (−0.156 − 0.888i)13-s + (1.30 + 2.26i)14-s + (−2.18 + 0.796i)15-s + (−0.939 + 0.342i)16-s + (0.126 − 0.719i)17-s + ⋯ |
L(s) = 1 | + (0.541 + 0.454i)2-s + (1.02 − 0.864i)3-s + (0.0868 + 0.492i)4-s + (−0.420 − 0.152i)5-s + 0.950·6-s + (0.926 + 0.337i)7-s + (−0.176 + 0.306i)8-s + (0.140 − 0.794i)9-s + (−0.158 − 0.273i)10-s + (0.285 − 0.494i)11-s + (0.514 + 0.432i)12-s + (−0.0434 − 0.246i)13-s + (0.348 + 0.604i)14-s + (−0.564 + 0.205i)15-s + (−0.234 + 0.0855i)16-s + (0.0307 − 0.174i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0301i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.41848 - 0.0364386i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.41848 - 0.0364386i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 5 | \( 1 + (0.939 + 0.342i)T \) |
| 37 | \( 1 + (1.62 - 5.86i)T \) |
good | 3 | \( 1 + (-1.78 + 1.49i)T + (0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (-2.45 - 0.892i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.947 + 1.64i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.156 + 0.888i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.126 + 0.719i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (3.60 - 3.02i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (0.507 + 0.878i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.79 + 3.11i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 5.42T + 31T^{2} \) |
| 41 | \( 1 + (1.19 + 6.76i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + 4.52T + 43T^{2} \) |
| 47 | \( 1 + (-2.52 - 4.37i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.31 - 2.29i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (2.99 - 1.09i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.39 - 13.5i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (5.82 + 2.11i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-3.08 + 2.58i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + 3.69T + 73T^{2} \) |
| 79 | \( 1 + (-5.57 - 2.02i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.504 + 2.85i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-2.41 + 0.879i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (4.45 + 7.71i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.73926366614934597654700593840, −10.65283435861240797567701985929, −9.031640503653718318052190402795, −8.293538165758619485976567640223, −7.81270052644826495299734946965, −6.79248889041744711974678254531, −5.60547422328103183400087030428, −4.35344950482049132472604435100, −3.11449101275951972469218514105, −1.80135731958188192620457014763,
1.98415295208640506950617574637, 3.35910355181060520090945020578, 4.24964466117240345017933949508, 4.97473023797430231957050061391, 6.67261984403403579057988387162, 7.83701882252447727998082643836, 8.775288868198557915969473873227, 9.602159911599338348966418442376, 10.61143739203742810583850219102, 11.24192001063988654271658581085