L(s) = 1 | − 5-s − 3·9-s − 11-s − 2·13-s − 17-s + 8·23-s + 25-s + 10·29-s − 6·37-s + 6·41-s + 4·43-s + 3·45-s − 12·47-s − 7·49-s + 6·53-s + 55-s + 12·59-s − 14·61-s + 2·65-s + 4·67-s − 8·71-s − 10·73-s − 4·79-s + 9·81-s + 12·83-s + 85-s + 10·89-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 9-s − 0.301·11-s − 0.554·13-s − 0.242·17-s + 1.66·23-s + 1/5·25-s + 1.85·29-s − 0.986·37-s + 0.937·41-s + 0.609·43-s + 0.447·45-s − 1.75·47-s − 49-s + 0.824·53-s + 0.134·55-s + 1.56·59-s − 1.79·61-s + 0.248·65-s + 0.488·67-s − 0.949·71-s − 1.17·73-s − 0.450·79-s + 81-s + 1.31·83-s + 0.108·85-s + 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.299188931\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.299188931\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.44528381439297, −13.80372894985955, −13.34271835570395, −12.73543263604851, −12.30044857882151, −11.69157592114654, −11.38823977532635, −10.69649019195240, −10.42103506533371, −9.634659571525876, −9.107245906077122, −8.561469358419126, −8.209977019100285, −7.546425967963912, −6.976505853570186, −6.497683587357192, −5.841291275048645, −5.092656647911396, −4.829143309078359, −4.122871301670008, −3.132145032606612, −2.980430456923651, −2.232564995543933, −1.222280745001860, −0.4137521444257337,
0.4137521444257337, 1.222280745001860, 2.232564995543933, 2.980430456923651, 3.132145032606612, 4.122871301670008, 4.829143309078359, 5.092656647911396, 5.841291275048645, 6.497683587357192, 6.976505853570186, 7.546425967963912, 8.209977019100285, 8.561469358419126, 9.107245906077122, 9.634659571525876, 10.42103506533371, 10.69649019195240, 11.38823977532635, 11.69157592114654, 12.30044857882151, 12.73543263604851, 13.34271835570395, 13.80372894985955, 14.44528381439297