| L(s) = 1 | + 1.18·2-s + 2.93·3-s − 0.589·4-s + 3.40·5-s + 3.48·6-s − 3.07·8-s + 5.61·9-s + 4.04·10-s − 3.26·11-s − 1.73·12-s + 9.98·15-s − 2.47·16-s + 2.55·17-s + 6.66·18-s + 4.96·19-s − 2.00·20-s − 3.88·22-s − 1.87·23-s − 9.02·24-s + 6.58·25-s + 7.66·27-s + 0.273·29-s + 11.8·30-s − 0.682·31-s + 3.21·32-s − 9.58·33-s + 3.03·34-s + ⋯ |
| L(s) = 1 | + 0.839·2-s + 1.69·3-s − 0.294·4-s + 1.52·5-s + 1.42·6-s − 1.08·8-s + 1.87·9-s + 1.27·10-s − 0.985·11-s − 0.499·12-s + 2.57·15-s − 0.618·16-s + 0.620·17-s + 1.57·18-s + 1.13·19-s − 0.449·20-s − 0.827·22-s − 0.391·23-s − 1.84·24-s + 1.31·25-s + 1.47·27-s + 0.0507·29-s + 2.16·30-s − 0.122·31-s + 0.568·32-s − 1.66·33-s + 0.521·34-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
| \(L(1)\) |
\(\approx\) |
\(7.092567079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.092567079\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.18T + 2T^{2} \) |
| 3 | \( 1 - 2.93T + 3T^{2} \) |
| 5 | \( 1 - 3.40T + 5T^{2} \) |
| 11 | \( 1 + 3.26T + 11T^{2} \) |
| 17 | \( 1 - 2.55T + 17T^{2} \) |
| 19 | \( 1 - 4.96T + 19T^{2} \) |
| 23 | \( 1 + 1.87T + 23T^{2} \) |
| 29 | \( 1 - 0.273T + 29T^{2} \) |
| 31 | \( 1 + 0.682T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 - 1.85T + 41T^{2} \) |
| 43 | \( 1 + 0.826T + 43T^{2} \) |
| 47 | \( 1 - 9.50T + 47T^{2} \) |
| 53 | \( 1 - 4.41T + 53T^{2} \) |
| 59 | \( 1 + 3.92T + 59T^{2} \) |
| 61 | \( 1 + 7.20T + 61T^{2} \) |
| 67 | \( 1 - 6.84T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 2.86T + 73T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 + 6.44T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 1.21T + 97T^{2} \) |
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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
−7.84328178492184446946767103416, −7.27668787188571654387088521585, −6.15188279927647142657551326733, −5.66440689511172746823715711697, −4.98385939273409850914066831734, −4.18697294644293512985732015671, −3.33400928000199387866004105770, −2.72055457358642227935342241974, −2.25562037101059959185263187285, −1.13028404135602317896382149702,
1.13028404135602317896382149702, 2.25562037101059959185263187285, 2.72055457358642227935342241974, 3.33400928000199387866004105770, 4.18697294644293512985732015671, 4.98385939273409850914066831734, 5.66440689511172746823715711697, 6.15188279927647142657551326733, 7.27668787188571654387088521585, 7.84328178492184446946767103416
