| L(s) = 1 | + 2-s − 4-s − 4·5-s − 7-s − 3·8-s − 4·10-s − 13-s − 14-s − 16-s − 17-s − 4·19-s + 4·20-s + 2·23-s + 11·25-s − 26-s + 28-s − 8·31-s + 5·32-s − 34-s + 4·35-s + 10·37-s − 4·38-s + 12·40-s + 4·43-s + 2·46-s + 6·47-s + 49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.78·5-s − 0.377·7-s − 1.06·8-s − 1.26·10-s − 0.277·13-s − 0.267·14-s − 1/4·16-s − 0.242·17-s − 0.917·19-s + 0.894·20-s + 0.417·23-s + 11/5·25-s − 0.196·26-s + 0.188·28-s − 1.43·31-s + 0.883·32-s − 0.171·34-s + 0.676·35-s + 1.64·37-s − 0.648·38-s + 1.89·40-s + 0.609·43-s + 0.294·46-s + 0.875·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13923 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13923 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−16.27660100423191, −15.79421927203551, −15.10963690127883, −14.81273185999893, −14.45424904625107, −13.42590352865928, −13.08539980994001, −12.46208015425868, −12.11666317880570, −11.48117665975069, −10.91577980779688, −10.32879267993853, −9.290766176133492, −8.974582675521306, −8.301867438947811, −7.656455251036263, −7.126886422304658, −6.369647682729828, −5.626096214828396, −4.858997087400315, −4.173877949346019, −3.933384512813963, −3.155233539710756, −2.426417013617885, −0.7981495960332012, 0,
0.7981495960332012, 2.426417013617885, 3.155233539710756, 3.933384512813963, 4.173877949346019, 4.858997087400315, 5.626096214828396, 6.369647682729828, 7.126886422304658, 7.656455251036263, 8.301867438947811, 8.974582675521306, 9.290766176133492, 10.32879267993853, 10.91577980779688, 11.48117665975069, 12.11666317880570, 12.46208015425868, 13.08539980994001, 13.42590352865928, 14.45424904625107, 14.81273185999893, 15.10963690127883, 15.79421927203551, 16.27660100423191
