L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.72 − 2.98i)5-s − 0.999·8-s − 3.44·10-s + (1 − 1.73i)11-s + (−2.44 − 4.24i)13-s + (−0.5 + 0.866i)16-s + 2·17-s − 7.44·19-s + (−1.72 + 2.98i)20-s + (−0.999 − 1.73i)22-s + (−0.5 − 0.866i)23-s + (−3.44 + 5.97i)25-s − 4.89·26-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.771 − 1.33i)5-s − 0.353·8-s − 1.09·10-s + (0.301 − 0.522i)11-s + (−0.679 − 1.17i)13-s + (−0.125 + 0.216i)16-s + 0.485·17-s − 1.70·19-s + (−0.385 + 0.667i)20-s + (−0.213 − 0.369i)22-s + (−0.104 − 0.180i)23-s + (−0.689 + 1.19i)25-s − 0.960·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6526353762\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6526353762\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.72 + 2.98i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.44 + 4.24i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 7.44T + 19T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.44 + 2.51i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 7.79T + 37T^{2} \) |
| 41 | \( 1 + (-4.89 - 8.48i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.44 + 2.51i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.89 + 8.48i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 1.10T + 53T^{2} \) |
| 59 | \( 1 + (-1 - 1.73i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.72 + 9.91i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.55 - 2.68i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 9.89T + 71T^{2} \) |
| 73 | \( 1 + 2.89T + 73T^{2} \) |
| 79 | \( 1 + (3.94 - 6.84i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1 - 1.73i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 7.10T + 89T^{2} \) |
| 97 | \( 1 + (3.44 - 5.97i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.456422693191094016505632035184, −7.85225694538208931205805378367, −6.71939565173105495520712960639, −5.69997338213311447469964492662, −5.02210347798528160777369122748, −4.30678077884985918323121468825, −3.55848355867994167579369832134, −2.49001447018966281721465186996, −1.15904113096762294727840466839, −0.20333762234735414125880753994,
2.07596865432345633931439009166, 2.98329565496159189623379596565, 4.12679973532163535094640997622, 4.36401956807987790191240908321, 5.72138455920183828080322760480, 6.54796287543131617900196837289, 7.08666808350214342163489339144, 7.55386890413707584400677132827, 8.496402641911257528549720691618, 9.263198160531840258785004434166