L(s) = 1 | + 2·3-s + 9-s − 4·13-s + 6·17-s + 2·19-s − 4·27-s − 6·29-s − 4·31-s − 2·37-s − 8·39-s − 6·41-s + 8·43-s + 12·47-s + 12·51-s − 6·53-s + 4·57-s − 6·59-s − 8·61-s − 4·67-s + 2·73-s − 8·79-s − 11·81-s + 6·83-s − 12·87-s + 6·89-s − 8·93-s − 10·97-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/3·9-s − 1.10·13-s + 1.45·17-s + 0.458·19-s − 0.769·27-s − 1.11·29-s − 0.718·31-s − 0.328·37-s − 1.28·39-s − 0.937·41-s + 1.21·43-s + 1.75·47-s + 1.68·51-s − 0.824·53-s + 0.529·57-s − 0.781·59-s − 1.02·61-s − 0.488·67-s + 0.234·73-s − 0.900·79-s − 1.22·81-s + 0.658·83-s − 1.28·87-s + 0.635·89-s − 0.829·93-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.83626678791664, −15.21683872481978, −14.82522073084538, −14.27846320853028, −13.97760469240791, −13.38487799536866, −12.61866661594068, −12.26493675159934, −11.67022441341907, −10.90340783707802, −10.28596718305103, −9.651587762341455, −9.252143583380024, −8.761409996962673, −7.918744941671745, −7.501994362884501, −7.244317753088007, −6.093982137199062, −5.527846220056572, −4.911073613114759, −3.984976732000847, −3.410046442824034, −2.815435577969025, −2.125830434823077, −1.299452261730195, 0,
1.299452261730195, 2.125830434823077, 2.815435577969025, 3.410046442824034, 3.984976732000847, 4.911073613114759, 5.527846220056572, 6.093982137199062, 7.244317753088007, 7.501994362884501, 7.918744941671745, 8.761409996962673, 9.252143583380024, 9.651587762341455, 10.28596718305103, 10.90340783707802, 11.67022441341907, 12.26493675159934, 12.61866661594068, 13.38487799536866, 13.97760469240791, 14.27846320853028, 14.82522073084538, 15.21683872481978, 15.83626678791664