| L(s) = 1 | + 7-s − 4·11-s + 6·13-s + 2·17-s − 6·19-s − 2·23-s − 6·29-s + 2·31-s + 4·37-s − 8·41-s − 4·43-s − 4·47-s + 49-s + 6·53-s + 4·59-s + 14·61-s + 4·67-s + 10·73-s − 4·77-s + 16·83-s − 8·89-s + 6·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.20·11-s + 1.66·13-s + 0.485·17-s − 1.37·19-s − 0.417·23-s − 1.11·29-s + 0.359·31-s + 0.657·37-s − 1.24·41-s − 0.609·43-s − 0.583·47-s + 1/7·49-s + 0.824·53-s + 0.520·59-s + 1.79·61-s + 0.488·67-s + 1.17·73-s − 0.455·77-s + 1.75·83-s − 0.847·89-s + 0.628·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.54764706424537, −15.08690157973177, −14.73084494886896, −13.84635069728964, −13.50198645839954, −12.96213666984459, −12.58918638748562, −11.70747902106444, −11.24530160534174, −10.78990080154218, −10.23174058680511, −9.748805997177532, −8.855070524084774, −8.238685289829354, −8.183844421401899, −7.309878465318254, −6.557206391817872, −6.064467695486387, −5.365810923774982, −4.903081878401357, −3.872578160116317, −3.649283823979504, −2.566309913920425, −1.965578210484576, −1.079894803656439, 0,
1.079894803656439, 1.965578210484576, 2.566309913920425, 3.649283823979504, 3.872578160116317, 4.903081878401357, 5.365810923774982, 6.064467695486387, 6.557206391817872, 7.309878465318254, 8.183844421401899, 8.238685289829354, 8.855070524084774, 9.748805997177532, 10.23174058680511, 10.78990080154218, 11.24530160534174, 11.70747902106444, 12.58918638748562, 12.96213666984459, 13.50198645839954, 13.84635069728964, 14.73084494886896, 15.08690157973177, 15.54764706424537
