L(s) = 1 | − 2·3-s + 9-s − 6·11-s + 2·13-s + 17-s − 4·19-s − 5·25-s + 4·27-s + 4·31-s + 12·33-s + 4·37-s − 4·39-s − 6·41-s − 8·43-s − 2·51-s + 6·53-s + 8·57-s − 4·61-s − 8·67-s − 2·73-s + 10·75-s + 8·79-s − 11·81-s + 6·89-s − 8·93-s − 14·97-s − 6·99-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/3·9-s − 1.80·11-s + 0.554·13-s + 0.242·17-s − 0.917·19-s − 25-s + 0.769·27-s + 0.718·31-s + 2.08·33-s + 0.657·37-s − 0.640·39-s − 0.937·41-s − 1.21·43-s − 0.280·51-s + 0.824·53-s + 1.05·57-s − 0.512·61-s − 0.977·67-s − 0.234·73-s + 1.15·75-s + 0.900·79-s − 1.22·81-s + 0.635·89-s − 0.829·93-s − 1.42·97-s − 0.603·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.86388560302725, −14.09337028347429, −13.45073880483394, −13.22440755714373, −12.63661125574016, −12.07395734312455, −11.59144138148356, −11.16196691972959, −10.53683020606456, −10.26320301257389, −9.838645102058224, −8.852348694823677, −8.417866511611559, −7.867283413873452, −7.363734512408603, −6.571340862687066, −6.082386861388870, −5.716569584399655, −4.977997234925646, −4.746851003745291, −3.859708650057943, −3.114301084678027, −2.442987704118711, −1.702342925604259, −0.6625528198484499, 0,
0.6625528198484499, 1.702342925604259, 2.442987704118711, 3.114301084678027, 3.859708650057943, 4.746851003745291, 4.977997234925646, 5.716569584399655, 6.082386861388870, 6.571340862687066, 7.363734512408603, 7.867283413873452, 8.417866511611559, 8.852348694823677, 9.838645102058224, 10.26320301257389, 10.53683020606456, 11.16196691972959, 11.59144138148356, 12.07395734312455, 12.63661125574016, 13.22440755714373, 13.45073880483394, 14.09337028347429, 14.86388560302725