L(s) = 1 | + (0.654 + 0.755i)2-s + (−0.959 + 0.281i)3-s + (−0.142 + 0.989i)4-s + (−0.841 − 0.540i)5-s + (−0.841 − 0.540i)6-s + (−0.841 + 0.540i)8-s + (0.841 − 0.540i)9-s + (−0.142 − 0.989i)10-s + (−0.142 − 0.989i)12-s + (0.959 + 0.281i)15-s + (−0.959 − 0.281i)16-s + (−1.14 + 0.989i)17-s + (0.959 + 0.281i)18-s + (−1.49 − 1.29i)19-s + (0.654 − 0.755i)20-s + ⋯ |
L(s) = 1 | + (0.654 + 0.755i)2-s + (−0.959 + 0.281i)3-s + (−0.142 + 0.989i)4-s + (−0.841 − 0.540i)5-s + (−0.841 − 0.540i)6-s + (−0.841 + 0.540i)8-s + (0.841 − 0.540i)9-s + (−0.142 − 0.989i)10-s + (−0.142 − 0.989i)12-s + (0.959 + 0.281i)15-s + (−0.959 − 0.281i)16-s + (−1.14 + 0.989i)17-s + (0.959 + 0.281i)18-s + (−1.49 − 1.29i)19-s + (0.654 − 0.755i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4175130890\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4175130890\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.654 - 0.755i)T \) |
| 3 | \( 1 + (0.959 - 0.281i)T \) |
| 5 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (-0.415 + 0.909i)T \) |
good | 7 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 11 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 13 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 17 | \( 1 + (1.14 - 0.989i)T + (0.142 - 0.989i)T^{2} \) |
| 19 | \( 1 + (1.49 + 1.29i)T + (0.142 + 0.989i)T^{2} \) |
| 29 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 31 | \( 1 + (-0.557 + 1.89i)T + (-0.841 - 0.540i)T^{2} \) |
| 37 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 41 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 43 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 47 | \( 1 + 1.81iT - T^{2} \) |
| 53 | \( 1 + (0.118 + 0.822i)T + (-0.959 + 0.281i)T^{2} \) |
| 59 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 61 | \( 1 + (-0.797 - 0.234i)T + (0.841 + 0.540i)T^{2} \) |
| 67 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 71 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 73 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 79 | \( 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2} \) |
| 83 | \( 1 + (0.304 + 0.474i)T + (-0.415 + 0.909i)T^{2} \) |
| 89 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 97 | \( 1 + (0.415 + 0.909i)T^{2} \) |
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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
−8.592542421135020901110077383522, −8.185999287347686317119968046078, −6.98002846710568467402103409262, −6.63705556562187626806738772709, −5.80645583960867705409323148118, −4.82763979017004591041000429664, −4.37346723837575963826319608142, −3.79666070918858233634608474769, −2.33110870170446155874688223449, −0.24554803152677288286510256279,
1.40345685381267789591088713420, 2.55979073289294652183496822029, 3.59087176408520470384905536635, 4.46531395735775383105670671558, 4.99503463639481285549114392882, 6.11288899471313756194736693737, 6.61104389948367595993435453275, 7.35319899682923704513571669649, 8.342019684230697865254013615270, 9.303435491418069778848328549030