| L(s) = 1 | + 1.52·2-s − 2.12·3-s + 0.312·4-s − 0.589·5-s − 3.23·6-s − 2.56·8-s + 1.52·9-s − 0.896·10-s − 1.52·11-s − 0.664·12-s + 1.25·15-s − 4.52·16-s + 4.79·17-s + 2.31·18-s − 1.68·19-s − 0.184·20-s − 2.31·22-s + 1.77·23-s + 5.45·24-s − 4.65·25-s + 3.14·27-s + 6.89·29-s + 1.90·30-s + 6.08·31-s − 1.75·32-s + 3.23·33-s + 7.29·34-s + ⋯ |
| L(s) = 1 | + 1.07·2-s − 1.22·3-s + 0.156·4-s − 0.263·5-s − 1.32·6-s − 0.907·8-s + 0.506·9-s − 0.283·10-s − 0.458·11-s − 0.191·12-s + 0.323·15-s − 1.13·16-s + 1.16·17-s + 0.545·18-s − 0.386·19-s − 0.0412·20-s − 0.493·22-s + 0.369·23-s + 1.11·24-s − 0.930·25-s + 0.605·27-s + 1.27·29-s + 0.347·30-s + 1.09·31-s − 0.310·32-s + 0.562·33-s + 1.25·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.52T + 2T^{2} \) |
| 3 | \( 1 + 2.12T + 3T^{2} \) |
| 5 | \( 1 + 0.589T + 5T^{2} \) |
| 11 | \( 1 + 1.52T + 11T^{2} \) |
| 17 | \( 1 - 4.79T + 17T^{2} \) |
| 19 | \( 1 + 1.68T + 19T^{2} \) |
| 23 | \( 1 - 1.77T + 23T^{2} \) |
| 29 | \( 1 - 6.89T + 29T^{2} \) |
| 31 | \( 1 - 6.08T + 31T^{2} \) |
| 37 | \( 1 + 1.40T + 37T^{2} \) |
| 41 | \( 1 + 1.35T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 0.464T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 2.48T + 61T^{2} \) |
| 67 | \( 1 - 7.57T + 67T^{2} \) |
| 71 | \( 1 + 6.60T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 - 0.973T + 97T^{2} \) |
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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
−7.05436312138060520180361610551, −6.58308838388530123020309072350, −5.77996744896745212920392526648, −5.39007694828148216289383228204, −4.79391552263049069350081814569, −4.10816728955854089215726209777, −3.29203663338605281611798135820, −2.51373308112053886567142665535, −1.03453572755467412603791511356, 0,
1.03453572755467412603791511356, 2.51373308112053886567142665535, 3.29203663338605281611798135820, 4.10816728955854089215726209777, 4.79391552263049069350081814569, 5.39007694828148216289383228204, 5.77996744896745212920392526648, 6.58308838388530123020309072350, 7.05436312138060520180361610551
