| L(s) = 1 | + 1.31·2-s + 1.82·3-s − 0.259·4-s − 4.11·5-s + 2.40·6-s − 2.74·7-s − 2.98·8-s + 0.329·9-s − 5.42·10-s + 4.56·11-s − 0.473·12-s − 6.36·13-s − 3.62·14-s − 7.50·15-s − 3.41·16-s − 3.25·17-s + 0.434·18-s + 6.59·19-s + 1.06·20-s − 5.01·21-s + 6.02·22-s + 23-s − 5.43·24-s + 11.9·25-s − 8.39·26-s − 4.87·27-s + 0.713·28-s + ⋯ |
| L(s) = 1 | + 0.932·2-s + 1.05·3-s − 0.129·4-s − 1.83·5-s + 0.982·6-s − 1.03·7-s − 1.05·8-s + 0.109·9-s − 1.71·10-s + 1.37·11-s − 0.136·12-s − 1.76·13-s − 0.969·14-s − 1.93·15-s − 0.853·16-s − 0.788·17-s + 0.102·18-s + 1.51·19-s + 0.238·20-s − 1.09·21-s + 1.28·22-s + 0.208·23-s − 1.11·24-s + 2.38·25-s − 1.64·26-s − 0.937·27-s + 0.134·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.31T + 2T^{2} \) |
| 3 | \( 1 - 1.82T + 3T^{2} \) |
| 5 | \( 1 + 4.11T + 5T^{2} \) |
| 7 | \( 1 + 2.74T + 7T^{2} \) |
| 11 | \( 1 - 4.56T + 11T^{2} \) |
| 13 | \( 1 + 6.36T + 13T^{2} \) |
| 17 | \( 1 + 3.25T + 17T^{2} \) |
| 19 | \( 1 - 6.59T + 19T^{2} \) |
| 31 | \( 1 + 9.08T + 31T^{2} \) |
| 37 | \( 1 - 6.61T + 37T^{2} \) |
| 41 | \( 1 + 2.61T + 41T^{2} \) |
| 43 | \( 1 + 8.65T + 43T^{2} \) |
| 47 | \( 1 + 5.78T + 47T^{2} \) |
| 53 | \( 1 - 1.33T + 53T^{2} \) |
| 59 | \( 1 - 7.66T + 59T^{2} \) |
| 61 | \( 1 - 5.22T + 61T^{2} \) |
| 67 | \( 1 - 1.97T + 67T^{2} \) |
| 71 | \( 1 + 4.65T + 71T^{2} \) |
| 73 | \( 1 + 6.95T + 73T^{2} \) |
| 79 | \( 1 - 9.90T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 2.94T + 89T^{2} \) |
| 97 | \( 1 - 2.26T + 97T^{2} \) |
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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
−9.684529591046766004183302971407, −9.229902899848233276157859240884, −8.372108134012971333732144511533, −7.35815541894135520105779360171, −6.72472585237385946944793832958, −5.17118422495876024526282011104, −4.14527707765167660772245914555, −3.50492017510678326787474296206, −2.85460522640893019783785320856, 0,
2.85460522640893019783785320856, 3.50492017510678326787474296206, 4.14527707765167660772245914555, 5.17118422495876024526282011104, 6.72472585237385946944793832958, 7.35815541894135520105779360171, 8.372108134012971333732144511533, 9.229902899848233276157859240884, 9.684529591046766004183302971407
