L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1 − 1.73i)5-s + (−2 + 3.46i)7-s − 0.999·8-s − 1.99·10-s + (2 − 3.46i)11-s + (−0.5 − 0.866i)13-s + (1.99 + 3.46i)14-s + (−0.5 + 0.866i)16-s + 2·17-s − 8·19-s + (−0.999 + 1.73i)20-s + (−1.99 − 3.46i)22-s + (0.500 − 0.866i)25-s − 0.999·26-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.447 − 0.774i)5-s + (−0.755 + 1.30i)7-s − 0.353·8-s − 0.632·10-s + (0.603 − 1.04i)11-s + (−0.138 − 0.240i)13-s + (0.534 + 0.925i)14-s + (−0.125 + 0.216i)16-s + 0.485·17-s − 1.83·19-s + (−0.223 + 0.387i)20-s + (−0.426 − 0.738i)22-s + (0.100 − 0.173i)25-s − 0.196·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4834786420\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4834786420\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (2 - 3.46i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + (-5 - 8.66i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8 - 13.8i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + (5 - 8.66i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.198914538413569305673129950575, −8.616597230179061777533022413734, −8.125527618259877548524029964704, −6.56939892921063731835185650909, −6.06766350416311642568376554146, −5.20958086036343826360680953489, −4.35635858891677468611969633572, −3.37116307495136432418963639261, −2.63490972280806157145185330029, −1.30065482510390929304246838235,
0.15454200569578258202671511413, 2.07991402950745603907448599566, 3.44414002986291886611269865326, 4.01186201556154754661209449785, 4.67809671467951452037535282676, 6.08945291311261957472758902633, 6.67692746492188835259554080275, 7.25723312310500697443651956905, 7.77093856022445267993631519569, 8.882204707245729639133251202750