| L(s) = 1 | − 4-s − 1.41·5-s + 9-s + 1.41·11-s + 16-s + 17-s + 1.41·20-s − 1.41·23-s + 1.00·25-s + 1.41·29-s + 1.41·31-s − 36-s − 1.41·41-s − 1.41·44-s − 1.41·45-s + 47-s + 49-s − 2·53-s − 2.00·55-s − 64-s − 68-s + 1.41·73-s − 1.41·80-s + 81-s − 2·83-s − 1.41·85-s − 2·89-s + ⋯ |
| L(s) = 1 | − 4-s − 1.41·5-s + 9-s + 1.41·11-s + 16-s + 17-s + 1.41·20-s − 1.41·23-s + 1.00·25-s + 1.41·29-s + 1.41·31-s − 36-s − 1.41·41-s − 1.41·44-s − 1.41·45-s + 47-s + 49-s − 2·53-s − 2.00·55-s − 64-s − 68-s + 1.41·73-s − 1.41·80-s + 81-s − 2·83-s − 1.41·85-s − 2·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7171353528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7171353528\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 1.41T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.41T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.41T + T^{2} \) |
| 29 | \( 1 - 1.41T + T^{2} \) |
| 31 | \( 1 - 1.41T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 53 | \( 1 + 2T + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.41T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 2T + T^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.20096756840628288540195937784, −9.746689803232361517317193832081, −8.601506233735390959462843882198, −8.063400185100257425644525510662, −7.17969365198483845132119138473, −6.16401959531337729925372970235, −4.70967097461339598712791931569, −4.13220800955407124516127928695, −3.42870379421018278691341708730, −1.14367168884042813850767301987,
1.14367168884042813850767301987, 3.42870379421018278691341708730, 4.13220800955407124516127928695, 4.70967097461339598712791931569, 6.16401959531337729925372970235, 7.17969365198483845132119138473, 8.063400185100257425644525510662, 8.601506233735390959462843882198, 9.746689803232361517317193832081, 10.20096756840628288540195937784
