L(s) = 1 | + 3·5-s − 11-s + 13-s − 4·17-s − 3·19-s − 6·23-s + 4·25-s + 5·29-s + 4·31-s − 37-s − 6·41-s + 4·43-s + 7·47-s + 2·53-s − 3·55-s + 59-s + 2·61-s + 3·65-s + 7·67-s + 10·71-s + 73-s + 16·79-s − 8·83-s − 12·85-s − 14·89-s − 9·95-s + 16·97-s + ⋯ |
L(s) = 1 | + 1.34·5-s − 0.301·11-s + 0.277·13-s − 0.970·17-s − 0.688·19-s − 1.25·23-s + 4/5·25-s + 0.928·29-s + 0.718·31-s − 0.164·37-s − 0.937·41-s + 0.609·43-s + 1.02·47-s + 0.274·53-s − 0.404·55-s + 0.130·59-s + 0.256·61-s + 0.372·65-s + 0.855·67-s + 1.18·71-s + 0.117·73-s + 1.80·79-s − 0.878·83-s − 1.30·85-s − 1.48·89-s − 0.923·95-s + 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.981122375\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.981122375\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−12.64584307307982, −12.34300545207300, −11.74030624181578, −11.23531194736396, −10.72062152563270, −10.25866329044815, −10.04659301427618, −9.480900551284983, −9.014018425499928, −8.505939041551141, −8.196657833572109, −7.578089609514718, −6.842088776484302, −6.504413254330790, −6.152816954353805, −5.617787066661567, −5.159515060382715, −4.576391988358033, −4.088860179293626, −3.508107985808636, −2.689259518900199, −2.257080274976085, −1.958200032656456, −1.173344315893428, −0.4553302749778841,
0.4553302749778841, 1.173344315893428, 1.958200032656456, 2.257080274976085, 2.689259518900199, 3.508107985808636, 4.088860179293626, 4.576391988358033, 5.159515060382715, 5.617787066661567, 6.152816954353805, 6.504413254330790, 6.842088776484302, 7.578089609514718, 8.196657833572109, 8.505939041551141, 9.014018425499928, 9.480900551284983, 10.04659301427618, 10.25866329044815, 10.72062152563270, 11.23531194736396, 11.74030624181578, 12.34300545207300, 12.64584307307982