| L(s) = 1 | + (−1.88 + 0.779i)2-s + (1.10 − 1.33i)3-s + (1.51 − 1.51i)4-s + (2.03 + 0.405i)5-s + (−1.03 + 3.37i)6-s + (0.388 + 1.95i)7-s + (−0.112 + 0.272i)8-s + (−0.565 − 2.94i)9-s + (−4.14 + 0.825i)10-s + (−1.30 + 1.95i)11-s + (−0.351 − 3.69i)12-s + (−4.24 − 4.24i)13-s + (−2.25 − 3.37i)14-s + (2.78 − 2.27i)15-s + 3.68i·16-s + (−4.01 + 0.958i)17-s + ⋯ |
| L(s) = 1 | + (−1.33 + 0.550i)2-s + (0.636 − 0.770i)3-s + (0.758 − 0.758i)4-s + (0.911 + 0.181i)5-s + (−0.422 + 1.37i)6-s + (0.146 + 0.738i)7-s + (−0.0399 + 0.0964i)8-s + (−0.188 − 0.982i)9-s + (−1.31 + 0.260i)10-s + (−0.394 + 0.590i)11-s + (−0.101 − 1.06i)12-s + (−1.17 − 1.17i)13-s + (−0.602 − 0.901i)14-s + (0.720 − 0.586i)15-s + 0.922i·16-s + (−0.972 + 0.232i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.589899 + 0.0487933i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.589899 + 0.0487933i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.10 + 1.33i)T \) |
| 17 | \( 1 + (4.01 - 0.958i)T \) |
| good | 2 | \( 1 + (1.88 - 0.779i)T + (1.41 - 1.41i)T^{2} \) |
| 5 | \( 1 + (-2.03 - 0.405i)T + (4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (-0.388 - 1.95i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (1.30 - 1.95i)T + (-4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (4.24 + 4.24i)T + 13iT^{2} \) |
| 19 | \( 1 + (-1.93 - 4.67i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.734 - 0.491i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (0.0116 - 0.0587i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (-2.44 + 1.63i)T + (11.8 - 28.6i)T^{2} \) |
| 37 | \( 1 + (1.52 + 2.28i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (-3.25 + 0.647i)T + (37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (2.53 - 6.12i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-5.76 + 5.76i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.75 + 0.725i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (3.54 - 8.56i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-5.30 + 1.05i)T + (56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + 7.09iT - 67T^{2} \) |
| 71 | \( 1 + (0.742 - 0.496i)T + (27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (0.551 - 2.77i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (3.30 + 2.20i)T + (30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-1.44 - 3.49i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-2.00 - 2.00i)T + 89iT^{2} \) |
| 97 | \( 1 + (-18.4 - 3.67i)T + (89.6 + 37.1i)T^{2} \) |
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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
−15.52182044245910987399588859313, −14.74806241949904808816322039420, −13.31488756473242273311728472119, −12.28533554844831267323624079538, −10.24959270780497712490154351909, −9.414514960084961719151333428362, −8.241695007736471587588949149635, −7.26871651164954847949210049383, −5.87226925951029180998174377879, −2.23752816047481829481073396164,
2.38205292699262167500618921286, 4.82683794113493126668902610446, 7.31204788974511473189345840157, 8.809776881697125029222141456277, 9.516866831665297774864919241468, 10.43525138537579699516467790503, 11.43363844597705669393988313676, 13.52886296772602355976133320716, 14.20842212222431042961277273296, 15.84839722683733996836241121932
