L(s) = 1 | + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (1 + 1.73i)9-s − 4·11-s + (−0.5 − 0.866i)13-s + (0.499 + 0.866i)15-s + (−1.5 + 2.59i)17-s + (−4 + 1.73i)19-s + (2.5 + 4.33i)23-s + (2 + 3.46i)25-s + 5·27-s + (3.5 + 6.06i)29-s − 4·31-s + (−2 + 3.46i)33-s − 10·37-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.223 + 0.387i)5-s + (0.333 + 0.577i)9-s − 1.20·11-s + (−0.138 − 0.240i)13-s + (0.129 + 0.223i)15-s + (−0.363 + 0.630i)17-s + (−0.917 + 0.397i)19-s + (0.521 + 0.902i)23-s + (0.400 + 0.692i)25-s + 0.962·27-s + (0.649 + 1.12i)29-s − 0.718·31-s + (−0.348 + 0.603i)33-s − 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.079394230\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.079394230\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (4 - 1.73i)T \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-2.5 - 4.33i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.5 - 6.06i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.5 + 4.33i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.5 + 6.06i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.5 - 9.52i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.5 - 9.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.5 + 9.52i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.5 - 12.9i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.30635681697216334422864240023, −8.857571436506208770288316013988, −8.325139298921436122740996078547, −7.30575442479180154324138690813, −7.04099428933538834886565069078, −5.67637069027213819610780671363, −4.93970543655673286845490453258, −3.69116018372694259187518944766, −2.66361088398248151877003270588, −1.64565947226456425102207556606,
0.42643685515750211111448020377, 2.29035348611369716291379545667, 3.28546026377520809595273889941, 4.53854471935038728507819800624, 4.86486881921466456229017161779, 6.26606043798695773688729472289, 7.00387469099120421651974817908, 8.108139705660945225610688135763, 8.690712882522587051949739378897, 9.513383454135585988270874709104