L(s) = 1 | − 3-s − 7-s + 9-s − 2·11-s + 2·13-s − 6·19-s + 21-s − 8·23-s − 27-s − 6·29-s − 10·31-s + 2·33-s + 8·37-s − 2·39-s − 6·41-s − 4·43-s − 8·47-s + 49-s − 2·53-s + 6·57-s − 12·59-s − 14·61-s − 63-s − 8·67-s + 8·69-s − 2·71-s − 2·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 1.37·19-s + 0.218·21-s − 1.66·23-s − 0.192·27-s − 1.11·29-s − 1.79·31-s + 0.348·33-s + 1.31·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.274·53-s + 0.794·57-s − 1.56·59-s − 1.79·61-s − 0.125·63-s − 0.977·67-s + 0.963·69-s − 0.237·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.09753381885\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09753381885\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−15.10573096689297, −14.59704902170279, −13.78896172119489, −13.37173733540823, −12.79269258725397, −12.51073808816381, −11.79892923804637, −11.21509321895149, −10.76591591697077, −10.32111682612252, −9.689019246946767, −9.151874570064533, −8.513465370564647, −7.802289263204236, −7.495965412992033, −6.541922585318076, −6.165524998175451, −5.721962585352456, −4.963768797583030, −4.299772189469860, −3.724795401597207, −3.039151719775317, −2.011376994187920, −1.595000810597640, −0.1156262937327781,
0.1156262937327781, 1.595000810597640, 2.011376994187920, 3.039151719775317, 3.724795401597207, 4.299772189469860, 4.963768797583030, 5.721962585352456, 6.165524998175451, 6.541922585318076, 7.495965412992033, 7.802289263204236, 8.513465370564647, 9.151874570064533, 9.689019246946767, 10.32111682612252, 10.76591591697077, 11.21509321895149, 11.79892923804637, 12.51073808816381, 12.79269258725397, 13.37173733540823, 13.78896172119489, 14.59704902170279, 15.10573096689297