L(s) = 1 | + (0.134 − 0.232i)2-s − 1.14·3-s + (0.964 + 1.66i)4-s + (1.28 + 2.21i)5-s + (−0.153 + 0.265i)6-s + (0.773 − 2.53i)7-s + 1.05·8-s − 1.69·9-s + 0.686·10-s + 3.94·11-s + (−1.10 − 1.90i)12-s + (−3.15 − 1.74i)13-s + (−0.483 − 0.518i)14-s + (−1.46 − 2.53i)15-s + (−1.78 + 3.09i)16-s + (−0.392 − 0.679i)17-s + ⋯ |
L(s) = 1 | + (0.0947 − 0.164i)2-s − 0.659·3-s + (0.482 + 0.834i)4-s + (0.572 + 0.992i)5-s + (−0.0625 + 0.108i)6-s + (0.292 − 0.956i)7-s + 0.372·8-s − 0.564·9-s + 0.217·10-s + 1.18·11-s + (−0.318 − 0.550i)12-s + (−0.874 − 0.484i)13-s + (−0.129 − 0.138i)14-s + (−0.378 − 0.654i)15-s + (−0.446 + 0.773i)16-s + (−0.0952 − 0.164i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.975834 + 0.222936i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.975834 + 0.222936i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.773 + 2.53i)T \) |
| 13 | \( 1 + (3.15 + 1.74i)T \) |
good | 2 | \( 1 + (-0.134 + 0.232i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + 1.14T + 3T^{2} \) |
| 5 | \( 1 + (-1.28 - 2.21i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 - 3.94T + 11T^{2} \) |
| 17 | \( 1 + (0.392 + 0.679i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + 7.49T + 19T^{2} \) |
| 23 | \( 1 + (-3.97 + 6.88i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.17 + 2.03i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.27 + 2.21i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.37 - 5.85i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.21 - 2.11i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.12 + 1.94i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.658 + 1.14i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.63 - 8.03i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.48 - 7.76i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 9.44T + 61T^{2} \) |
| 67 | \( 1 + 1.35T + 67T^{2} \) |
| 71 | \( 1 + (6.15 - 10.6i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.384 - 0.665i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.09 + 5.36i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 1.07T + 83T^{2} \) |
| 89 | \( 1 + (3.83 - 6.63i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.18 + 2.05i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−14.25665149687092050893091800133, −12.98612336989371191517653654250, −11.88600218551350098302009828527, −10.96517216362457356249000421450, −10.30812704168354523948377283050, −8.496431533441518691760817241402, −7.00575749009347387700981669530, −6.37067939331133700901296559288, −4.35703602722561885347661362994, −2.66162183724471948812018366829,
1.83464653610401243271167157870, 4.89389388606952577196881471963, 5.70213113438885450317625325141, 6.71360741457434206127121291665, 8.761773961348950682193380237474, 9.498922178270736048861091683354, 11.01451701721186623279864227321, 11.82124610428359995933448855516, 12.78241516764902391913890075140, 14.29610007424518368562952607293