| L(s) = 1 | + 3·5-s − 6·11-s + 2·13-s + 7·17-s − 2·19-s − 4·23-s + 4·25-s − 2·29-s − 2·31-s − 3·37-s − 3·41-s − 11·43-s + 7·47-s − 12·53-s − 18·55-s + 3·59-s − 10·61-s + 6·65-s − 4·67-s − 12·71-s + 16·73-s − 7·79-s − 7·83-s + 21·85-s + 10·89-s − 6·95-s + 2·97-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.80·11-s + 0.554·13-s + 1.69·17-s − 0.458·19-s − 0.834·23-s + 4/5·25-s − 0.371·29-s − 0.359·31-s − 0.493·37-s − 0.468·41-s − 1.67·43-s + 1.02·47-s − 1.64·53-s − 2.42·55-s + 0.390·59-s − 1.28·61-s + 0.744·65-s − 0.488·67-s − 1.42·71-s + 1.87·73-s − 0.787·79-s − 0.768·83-s + 2.27·85-s + 1.05·89-s − 0.615·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.87629940163763, −16.34849090284381, −15.69432751741448, −15.20710752920649, −14.33418773900392, −14.03696087785028, −13.27911846281448, −13.05590826791104, −12.34375237809625, −11.70062024022519, −10.70813302022718, −10.40799624152854, −9.931698481767995, −9.342059578037278, −8.499185415930655, −7.932396495894218, −7.384739971691743, −6.388339649440520, −5.827772750675247, −5.377846776585182, −4.778462866514482, −3.587497281056252, −2.933410351982093, −2.083647895464070, −1.420364443039087, 0,
1.420364443039087, 2.083647895464070, 2.933410351982093, 3.587497281056252, 4.778462866514482, 5.377846776585182, 5.827772750675247, 6.388339649440520, 7.384739971691743, 7.932396495894218, 8.499185415930655, 9.342059578037278, 9.931698481767995, 10.40799624152854, 10.70813302022718, 11.70062024022519, 12.34375237809625, 13.05590826791104, 13.27911846281448, 14.03696087785028, 14.33418773900392, 15.20710752920649, 15.69432751741448, 16.34849090284381, 16.87629940163763
