L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.500 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−1 − 1.73i)7-s − 3·8-s + 0.999·10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + (−0.999 + 1.73i)14-s + (0.500 + 0.866i)16-s − 2·17-s − 3·19-s + (0.499 + 0.866i)20-s + (−0.499 + 0.866i)22-s + (−0.499 − 0.866i)25-s − 0.999·26-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.250 − 0.433i)4-s + (−0.223 + 0.387i)5-s + (−0.377 − 0.654i)7-s − 1.06·8-s + 0.316·10-s + (−0.150 − 0.261i)11-s + (0.138 − 0.240i)13-s + (−0.267 + 0.462i)14-s + (0.125 + 0.216i)16-s − 0.485·17-s − 0.688·19-s + (0.111 + 0.193i)20-s + (−0.106 + 0.184i)22-s + (−0.0999 − 0.173i)25-s − 0.196·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 14T + 53T^{2} \) |
| 59 | \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5 + 8.66i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.894099906779463724199322950964, −8.065496853768385230930604172136, −7.00117035277627246441829788683, −6.46163334418971755662590960971, −5.58341290204056281625952393212, −4.42325939299105524856234472394, −3.37463358066403399753786183922, −2.57977611524376115471689126586, −1.32547714595512309609106092183, 0,
2.03028915539389220282682487604, 3.03091185132671330206734504038, 4.07926188039614167590725058724, 5.09667498073455144308057212363, 6.17705086673578182753206858978, 6.62093839365958432014770772621, 7.63916520216298262106241334035, 8.240372061130828114126407554162, 9.008135041586029053512543974070