| L(s) = 1 | + 4·11-s − 13-s − 6·17-s − 4·19-s − 8·23-s − 6·29-s + 8·31-s + 10·37-s + 6·41-s + 4·43-s − 7·49-s − 10·53-s + 4·59-s − 2·61-s − 12·67-s + 16·71-s − 2·73-s + 16·79-s + 12·83-s − 10·89-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s − 0.277·13-s − 1.45·17-s − 0.917·19-s − 1.66·23-s − 1.11·29-s + 1.43·31-s + 1.64·37-s + 0.937·41-s + 0.609·43-s − 49-s − 1.37·53-s + 0.520·59-s − 0.256·61-s − 1.46·67-s + 1.89·71-s − 0.234·73-s + 1.80·79-s + 1.31·83-s − 1.05·89-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.78832878762838, −14.42888701941910, −13.78322402565867, −13.42160113372346, −12.69199576942024, −12.37263305763015, −11.70169854411814, −11.21648667788208, −10.88879074432519, −10.10071993750566, −9.486144372496381, −9.267928596896295, −8.527809798974710, −7.997785713447042, −7.516638935884457, −6.639953980293518, −6.275861533383546, −5.991815607889554, −4.920834797398042, −4.281297406804537, −4.108908804395569, −3.227432088994946, −2.263654019418552, −1.981491861840796, −0.9297917277576514, 0,
0.9297917277576514, 1.981491861840796, 2.263654019418552, 3.227432088994946, 4.108908804395569, 4.281297406804537, 4.920834797398042, 5.991815607889554, 6.275861533383546, 6.639953980293518, 7.516638935884457, 7.997785713447042, 8.527809798974710, 9.267928596896295, 9.486144372496381, 10.10071993750566, 10.88879074432519, 11.21648667788208, 11.70169854411814, 12.37263305763015, 12.69199576942024, 13.42160113372346, 13.78322402565867, 14.42888701941910, 14.78832878762838
