L(s) = 1 | + 5-s + 7-s − 0.449·11-s − 2.44·13-s − 1.44·17-s − 7.34·19-s − 23-s + 25-s + 4.55·29-s + 7.89·31-s + 35-s − 0.101·37-s + 5.44·41-s + 11.7·43-s + 7.34·47-s − 6·49-s + 10.3·53-s − 0.449·55-s + 11.4·59-s − 4.44·61-s − 2.44·65-s − 10.7·67-s + 12.3·71-s + 7.34·73-s − 0.449·77-s + 5.79·79-s + 10.5·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.135·11-s − 0.679·13-s − 0.351·17-s − 1.68·19-s − 0.208·23-s + 0.200·25-s + 0.845·29-s + 1.41·31-s + 0.169·35-s − 0.0166·37-s + 0.851·41-s + 1.79·43-s + 1.07·47-s − 0.857·49-s + 1.42·53-s − 0.0606·55-s + 1.49·59-s − 0.569·61-s − 0.303·65-s − 1.31·67-s + 1.46·71-s + 0.860·73-s − 0.0512·77-s + 0.652·79-s + 1.15·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.976329105\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.976329105\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 0.449T + 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 + 1.44T + 17T^{2} \) |
| 19 | \( 1 + 7.34T + 19T^{2} \) |
| 29 | \( 1 - 4.55T + 29T^{2} \) |
| 31 | \( 1 - 7.89T + 31T^{2} \) |
| 37 | \( 1 + 0.101T + 37T^{2} \) |
| 41 | \( 1 - 5.44T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 7.34T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 4.44T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 7.34T + 73T^{2} \) |
| 79 | \( 1 - 5.79T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 4.89T + 97T^{2} \) |
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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
−8.378843067517269923985759060009, −7.80319161465039942699747762390, −6.83920302859602897722186756324, −6.28376803028986488574203348391, −5.45514262545136575658780105709, −4.59226438787871234996184266399, −4.04682575812786842521816566539, −2.62722058986211987942536223678, −2.17984878007080163743100052573, −0.792486365317966618573500168079,
0.792486365317966618573500168079, 2.17984878007080163743100052573, 2.62722058986211987942536223678, 4.04682575812786842521816566539, 4.59226438787871234996184266399, 5.45514262545136575658780105709, 6.28376803028986488574203348391, 6.83920302859602897722186756324, 7.80319161465039942699747762390, 8.378843067517269923985759060009