| L(s) = 1 | + 1.35·2-s − 1.32·3-s − 0.167·4-s + 1.38·5-s − 1.79·6-s − 1.97·7-s − 2.93·8-s − 1.23·9-s + 1.87·10-s + 3.53·11-s + 0.222·12-s − 2.23·13-s − 2.67·14-s − 1.84·15-s − 3.63·16-s − 2.50·17-s − 1.67·18-s − 8.35·19-s − 0.232·20-s + 2.62·21-s + 4.78·22-s + 23-s + 3.89·24-s − 3.07·25-s − 3.02·26-s + 5.62·27-s + 0.330·28-s + ⋯ |
| L(s) = 1 | + 0.957·2-s − 0.767·3-s − 0.0837·4-s + 0.619·5-s − 0.734·6-s − 0.746·7-s − 1.03·8-s − 0.411·9-s + 0.593·10-s + 1.06·11-s + 0.0642·12-s − 0.619·13-s − 0.714·14-s − 0.475·15-s − 0.909·16-s − 0.606·17-s − 0.393·18-s − 1.91·19-s − 0.0519·20-s + 0.573·21-s + 1.02·22-s + 0.208·23-s + 0.795·24-s − 0.615·25-s − 0.593·26-s + 1.08·27-s + 0.0625·28-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 | \( 1 - 1.35T + 2T^{2} \) |
| 3 | \( 1 + 1.32T + 3T^{2} \) |
| 5 | \( 1 - 1.38T + 5T^{2} \) |
| 7 | \( 1 + 1.97T + 7T^{2} \) |
| 11 | \( 1 - 3.53T + 11T^{2} \) |
| 13 | \( 1 + 2.23T + 13T^{2} \) |
| 17 | \( 1 + 2.50T + 17T^{2} \) |
| 19 | \( 1 + 8.35T + 19T^{2} \) |
| 31 | \( 1 + 4.77T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 + 7.41T + 41T^{2} \) |
| 43 | \( 1 - 0.277T + 43T^{2} \) |
| 47 | \( 1 - 13.6T + 47T^{2} \) |
| 53 | \( 1 - 0.801T + 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 - 0.569T + 61T^{2} \) |
| 67 | \( 1 + 1.69T + 67T^{2} \) |
| 71 | \( 1 - 6.09T + 71T^{2} \) |
| 73 | \( 1 + 3.10T + 73T^{2} \) |
| 79 | \( 1 - 2.74T + 79T^{2} \) |
| 83 | \( 1 + 1.75T + 83T^{2} \) |
| 89 | \( 1 - 6.56T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−10.17732677890222521392860506222, −9.204715292284599885274696667988, −8.637874234418204607888345794525, −6.82444661161790620123432225905, −6.27970097644433500925891993714, −5.58513591880290462445972365108, −4.58926449203226554748358790435, −3.65409939818002221339829706959, −2.31088787554856910767080071574, 0,
2.31088787554856910767080071574, 3.65409939818002221339829706959, 4.58926449203226554748358790435, 5.58513591880290462445972365108, 6.27970097644433500925891993714, 6.82444661161790620123432225905, 8.637874234418204607888345794525, 9.204715292284599885274696667988, 10.17732677890222521392860506222
