| L(s) = 1 | + 0.913·2-s + 0.436·3-s − 1.16·4-s − 0.348·5-s + 0.398·6-s + 2.49·7-s − 2.89·8-s − 2.80·9-s − 0.318·10-s + 1.52·11-s − 0.508·12-s − 1.93·13-s + 2.28·14-s − 0.152·15-s − 0.311·16-s − 1.84·17-s − 2.56·18-s − 6.66·19-s + 0.406·20-s + 1.08·21-s + 1.39·22-s + 5.27·23-s − 1.26·24-s − 4.87·25-s − 1.76·26-s − 2.53·27-s − 2.90·28-s + ⋯ |
| L(s) = 1 | + 0.646·2-s + 0.251·3-s − 0.582·4-s − 0.155·5-s + 0.162·6-s + 0.943·7-s − 1.02·8-s − 0.936·9-s − 0.100·10-s + 0.458·11-s − 0.146·12-s − 0.535·13-s + 0.609·14-s − 0.0392·15-s − 0.0778·16-s − 0.447·17-s − 0.605·18-s − 1.52·19-s + 0.0908·20-s + 0.237·21-s + 0.296·22-s + 1.10·23-s − 0.257·24-s − 0.975·25-s − 0.345·26-s − 0.487·27-s − 0.549·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 1511 | \( 1 + T \) |
| good | 2 | \( 1 - 0.913T + 2T^{2} \) |
| 3 | \( 1 - 0.436T + 3T^{2} \) |
| 5 | \( 1 + 0.348T + 5T^{2} \) |
| 7 | \( 1 - 2.49T + 7T^{2} \) |
| 11 | \( 1 - 1.52T + 11T^{2} \) |
| 13 | \( 1 + 1.93T + 13T^{2} \) |
| 17 | \( 1 + 1.84T + 17T^{2} \) |
| 19 | \( 1 + 6.66T + 19T^{2} \) |
| 23 | \( 1 - 5.27T + 23T^{2} \) |
| 29 | \( 1 + 8.85T + 29T^{2} \) |
| 31 | \( 1 + 2.36T + 31T^{2} \) |
| 37 | \( 1 - 7.99T + 37T^{2} \) |
| 41 | \( 1 - 4.21T + 41T^{2} \) |
| 43 | \( 1 - 1.94T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 + 9.24T + 53T^{2} \) |
| 59 | \( 1 + 4.06T + 59T^{2} \) |
| 61 | \( 1 - 5.98T + 61T^{2} \) |
| 67 | \( 1 + 15.7T + 67T^{2} \) |
| 71 | \( 1 + 9.49T + 71T^{2} \) |
| 73 | \( 1 - 7.53T + 73T^{2} \) |
| 79 | \( 1 + 1.56T + 79T^{2} \) |
| 83 | \( 1 - 6.96T + 83T^{2} \) |
| 89 | \( 1 + 0.0127T + 89T^{2} \) |
| 97 | \( 1 + 1.29T + 97T^{2} \) |
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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
−9.075130252564269328533619983765, −8.298393896640279230920963095750, −7.65614487266416786164494992264, −6.39604769602915997579008499924, −5.64037721007013894379190877845, −4.74227977469366351532186194339, −4.12199606778224787528007644826, −3.08164786356662830329395731336, −1.93914089235034105094811692007, 0,
1.93914089235034105094811692007, 3.08164786356662830329395731336, 4.12199606778224787528007644826, 4.74227977469366351532186194339, 5.64037721007013894379190877845, 6.39604769602915997579008499924, 7.65614487266416786164494992264, 8.298393896640279230920963095750, 9.075130252564269328533619983765
