| L(s) = 1 | + 2.69·2-s + 5.28·4-s + 1.58·5-s + 8.87·8-s + 4.28·10-s + 0.300·11-s − 2.81·13-s + 13.3·16-s − 5.87·17-s + 2.28·19-s + 8.39·20-s + 0.810·22-s + 1.88·23-s − 2.47·25-s − 7.58·26-s + 2.52·29-s + 4.81·31-s + 18.3·32-s − 15.8·34-s − 4.47·37-s + 6.17·38-s + 14.0·40-s − 8.90·41-s − 9.09·43-s + 1.58·44-s + 5.09·46-s + 3.21·47-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 2.64·4-s + 0.710·5-s + 3.13·8-s + 1.35·10-s + 0.0905·11-s − 0.779·13-s + 3.34·16-s − 1.42·17-s + 0.524·19-s + 1.87·20-s + 0.172·22-s + 0.393·23-s − 0.495·25-s − 1.48·26-s + 0.468·29-s + 0.864·31-s + 3.25·32-s − 2.72·34-s − 0.736·37-s + 1.00·38-s + 2.22·40-s − 1.39·41-s − 1.38·43-s + 0.239·44-s + 0.751·46-s + 0.468·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.963198269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.963198269\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 5 | \( 1 - 1.58T + 5T^{2} \) |
| 11 | \( 1 - 0.300T + 11T^{2} \) |
| 13 | \( 1 + 2.81T + 13T^{2} \) |
| 17 | \( 1 + 5.87T + 17T^{2} \) |
| 19 | \( 1 - 2.28T + 19T^{2} \) |
| 23 | \( 1 - 1.88T + 23T^{2} \) |
| 29 | \( 1 - 2.52T + 29T^{2} \) |
| 31 | \( 1 - 4.81T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 8.90T + 41T^{2} \) |
| 43 | \( 1 + 9.09T + 43T^{2} \) |
| 47 | \( 1 - 3.21T + 47T^{2} \) |
| 53 | \( 1 - 2.01T + 53T^{2} \) |
| 59 | \( 1 - 4.88T + 59T^{2} \) |
| 61 | \( 1 + 7.57T + 61T^{2} \) |
| 67 | \( 1 - 0.712T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 1.66T + 79T^{2} \) |
| 83 | \( 1 - 5.43T + 83T^{2} \) |
| 89 | \( 1 - 9.35T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−9.973151888900098123937945500351, −8.799004814741339864826210230952, −7.57957284100525440084382724710, −6.72826351805603779380876472696, −6.23206650799529291774439212010, −5.16386850030209061018530219915, −4.73894802588754734597392581667, −3.63177191814584300706651281048, −2.63582510607946938269152046297, −1.81289808073203684792573398015,
1.81289808073203684792573398015, 2.63582510607946938269152046297, 3.63177191814584300706651281048, 4.73894802588754734597392581667, 5.16386850030209061018530219915, 6.23206650799529291774439212010, 6.72826351805603779380876472696, 7.57957284100525440084382724710, 8.799004814741339864826210230952, 9.973151888900098123937945500351
