| L(s) = 1 | + 1.77·2-s + 0.618·3-s + 1.15·4-s − 2.77·5-s + 1.09·6-s − 7-s − 1.49·8-s − 2.61·9-s − 4.93·10-s + 0.716·12-s − 4.29·13-s − 1.77·14-s − 1.71·15-s − 4.97·16-s − 2.75·17-s − 4.65·18-s + 1.93·19-s − 3.22·20-s − 0.618·21-s + 4.37·23-s − 0.923·24-s + 2.71·25-s − 7.63·26-s − 3.47·27-s − 1.15·28-s + 8.62·29-s − 3.05·30-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 0.356·3-s + 0.579·4-s − 1.24·5-s + 0.448·6-s − 0.377·7-s − 0.528·8-s − 0.872·9-s − 1.56·10-s + 0.206·12-s − 1.19·13-s − 0.475·14-s − 0.443·15-s − 1.24·16-s − 0.668·17-s − 1.09·18-s + 0.444·19-s − 0.720·20-s − 0.134·21-s + 0.911·23-s − 0.188·24-s + 0.542·25-s − 1.49·26-s − 0.668·27-s − 0.219·28-s + 1.60·29-s − 0.557·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.77T + 2T^{2} \) |
| 3 | \( 1 - 0.618T + 3T^{2} \) |
| 5 | \( 1 + 2.77T + 5T^{2} \) |
| 13 | \( 1 + 4.29T + 13T^{2} \) |
| 17 | \( 1 + 2.75T + 17T^{2} \) |
| 19 | \( 1 - 1.93T + 19T^{2} \) |
| 23 | \( 1 - 4.37T + 23T^{2} \) |
| 29 | \( 1 - 8.62T + 29T^{2} \) |
| 31 | \( 1 + 0.200T + 31T^{2} \) |
| 37 | \( 1 - 1.03T + 37T^{2} \) |
| 41 | \( 1 - 9.60T + 41T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 + 13.0T + 47T^{2} \) |
| 53 | \( 1 + 3.90T + 53T^{2} \) |
| 59 | \( 1 + 8.55T + 59T^{2} \) |
| 61 | \( 1 + 0.988T + 61T^{2} \) |
| 67 | \( 1 + 5.41T + 67T^{2} \) |
| 71 | \( 1 + 2.01T + 71T^{2} \) |
| 73 | \( 1 + 9.97T + 73T^{2} \) |
| 79 | \( 1 - 6.29T + 79T^{2} \) |
| 83 | \( 1 + 1.72T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−9.673579831401046306610646814365, −8.845631215769053853261025126617, −7.993850797990009419175843052472, −7.07020885529818200153623587940, −6.16002940221313045693049134577, −4.99288868020271546047632544258, −4.38942106381888037765767415338, −3.25712326342724889913944253506, −2.72754363924799721330696989366, 0,
2.72754363924799721330696989366, 3.25712326342724889913944253506, 4.38942106381888037765767415338, 4.99288868020271546047632544258, 6.16002940221313045693049134577, 7.07020885529818200153623587940, 7.993850797990009419175843052472, 8.845631215769053853261025126617, 9.673579831401046306610646814365
