L(s) = 1 | + 2.57·2-s − 0.743·3-s + 4.64·4-s + 3.77·5-s − 1.91·6-s + 6.81·8-s − 2.44·9-s + 9.73·10-s + 5.38·11-s − 3.45·12-s − 2.81·15-s + 8.28·16-s − 6.31·17-s − 6.30·18-s + 3.32·19-s + 17.5·20-s + 13.8·22-s − 1.01·23-s − 5.06·24-s + 9.27·25-s + 4.05·27-s + 0.452·29-s − 7.24·30-s + 0.481·31-s + 7.71·32-s − 4.00·33-s − 16.2·34-s + ⋯ |
L(s) = 1 | + 1.82·2-s − 0.429·3-s + 2.32·4-s + 1.68·5-s − 0.782·6-s + 2.40·8-s − 0.815·9-s + 3.07·10-s + 1.62·11-s − 0.997·12-s − 0.725·15-s + 2.07·16-s − 1.53·17-s − 1.48·18-s + 0.762·19-s + 3.92·20-s + 2.96·22-s − 0.210·23-s − 1.03·24-s + 1.85·25-s + 0.779·27-s + 0.0839·29-s − 1.32·30-s + 0.0865·31-s + 1.36·32-s − 0.697·33-s − 2.79·34-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
\(L(1)\) |
\(\approx\) |
\(8.708765697\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.708765697\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.57T + 2T^{2} \) |
| 3 | \( 1 + 0.743T + 3T^{2} \) |
| 5 | \( 1 - 3.77T + 5T^{2} \) |
| 11 | \( 1 - 5.38T + 11T^{2} \) |
| 17 | \( 1 + 6.31T + 17T^{2} \) |
| 19 | \( 1 - 3.32T + 19T^{2} \) |
| 23 | \( 1 + 1.01T + 23T^{2} \) |
| 29 | \( 1 - 0.452T + 29T^{2} \) |
| 31 | \( 1 - 0.481T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 + 0.112T + 41T^{2} \) |
| 43 | \( 1 + 2.93T + 43T^{2} \) |
| 47 | \( 1 + 1.19T + 47T^{2} \) |
| 53 | \( 1 - 2.33T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 4.03T + 61T^{2} \) |
| 67 | \( 1 + 0.704T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 15.8T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + 0.284T + 97T^{2} \) |
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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
−7.28271826589858295929833362805, −6.50738050021745462170794561764, −6.25038493515203686643293193922, −5.80824032737704515871208939418, −5.01449045183559033835885562883, −4.49880427597822305815170459734, −3.60015543281331357902870817444, −2.71739894012088261884535426047, −2.11443646446023077973938633782, −1.23144478908288706296204304900,
1.23144478908288706296204304900, 2.11443646446023077973938633782, 2.71739894012088261884535426047, 3.60015543281331357902870817444, 4.49880427597822305815170459734, 5.01449045183559033835885562883, 5.80824032737704515871208939418, 6.25038493515203686643293193922, 6.50738050021745462170794561764, 7.28271826589858295929833362805