| L(s) = 1 | − 3·9-s − 11-s − 2·13-s − 4·17-s − 2·19-s − 5·23-s + 29-s − 2·31-s + 3·37-s − 12·41-s − 11·43-s + 2·47-s + 6·53-s − 10·59-s − 4·61-s − 67-s + 3·71-s + 9·79-s + 9·81-s − 2·83-s + 6·89-s − 14·97-s + 3·99-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 9-s − 0.301·11-s − 0.554·13-s − 0.970·17-s − 0.458·19-s − 1.04·23-s + 0.185·29-s − 0.359·31-s + 0.493·37-s − 1.87·41-s − 1.67·43-s + 0.291·47-s + 0.824·53-s − 1.30·59-s − 0.512·61-s − 0.122·67-s + 0.356·71-s + 1.01·79-s + 81-s − 0.219·83-s + 0.635·89-s − 1.42·97-s + 0.301·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5849612007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5849612007\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.55468395038117, −15.13055794434337, −14.69798853455436, −13.96052277069671, −13.57436501913294, −13.06491968481786, −12.21654391568701, −11.92844630033050, −11.28098254392406, −10.71839186243920, −10.16418390050140, −9.551672929733957, −8.857088363834766, −8.370128690824530, −7.890718551129548, −7.078149431079496, −6.470172869278956, −5.931229738523224, −5.157266801259753, −4.677153413486846, −3.824374912656717, −3.103288653125651, −2.353553278884454, −1.739900913037666, −0.2965779108152829,
0.2965779108152829, 1.739900913037666, 2.353553278884454, 3.103288653125651, 3.824374912656717, 4.677153413486846, 5.157266801259753, 5.931229738523224, 6.470172869278956, 7.078149431079496, 7.890718551129548, 8.370128690824530, 8.857088363834766, 9.551672929733957, 10.16418390050140, 10.71839186243920, 11.28098254392406, 11.92844630033050, 12.21654391568701, 13.06491968481786, 13.57436501913294, 13.96052277069671, 14.69798853455436, 15.13055794434337, 15.55468395038117
