L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 5-s + (2 − 1.73i)7-s + 0.999·8-s + (−0.5 + 0.866i)10-s + 5·11-s + (0.499 + 2.59i)14-s + (−0.5 + 0.866i)16-s + (2 − 3.46i)17-s + (−4 − 6.92i)19-s + (−0.499 − 0.866i)20-s + (−2.5 + 4.33i)22-s − 4·23-s − 4·25-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.447·5-s + (0.755 − 0.654i)7-s + 0.353·8-s + (−0.158 + 0.273i)10-s + 1.50·11-s + (0.133 + 0.694i)14-s + (−0.125 + 0.216i)16-s + (0.485 − 0.840i)17-s + (−0.917 − 1.58i)19-s + (−0.111 − 0.193i)20-s + (−0.533 + 0.923i)22-s − 0.834·23-s − 0.800·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.612144843\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.612144843\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2 + 1.73i)T \) |
good | 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.5 - 9.52i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.5 - 2.59i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.5 - 6.06i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.5 + 6.06i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.668074273613551459688038262444, −8.917908129767219229642457217633, −8.206947165046150323438989344873, −7.12022006626006840660572310824, −6.69597695111171852071264192433, −5.62675121393736725652141129291, −4.67272674170603994530873488218, −3.84287671461810534205376915146, −2.13974118550436682004688967592, −0.889780786249069642587307314762,
1.47261802301261065866000371488, 2.13762194134070572957538737961, 3.68862911514877503112798619708, 4.36105895330775565128051570096, 5.80513449795066747172927171629, 6.24613804831363455531798114053, 7.69186550839023917050464081808, 8.353306665935303634452004410265, 9.077556515309821629133765072707, 9.885091742322668318154652261159