L(s) = 1 | − 2·3-s + 7-s + 9-s − 2·13-s + 6·17-s + 4·19-s − 2·21-s − 23-s + 4·27-s + 6·29-s − 2·31-s + 10·37-s + 4·39-s − 6·41-s − 10·43-s + 49-s − 12·51-s − 6·53-s − 8·57-s + 12·59-s − 10·61-s + 63-s + 2·67-s + 2·69-s − 12·71-s + 16·73-s + 10·79-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.554·13-s + 1.45·17-s + 0.917·19-s − 0.436·21-s − 0.208·23-s + 0.769·27-s + 1.11·29-s − 0.359·31-s + 1.64·37-s + 0.640·39-s − 0.937·41-s − 1.52·43-s + 1/7·49-s − 1.68·51-s − 0.824·53-s − 1.05·57-s + 1.56·59-s − 1.28·61-s + 0.125·63-s + 0.244·67-s + 0.240·69-s − 1.42·71-s + 1.87·73-s + 1.12·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−14.38775632971249, −14.09281108965422, −13.49623325892344, −12.76588524091293, −12.34994123324681, −11.88861666616304, −11.49947368347028, −11.15472713190597, −10.33151086759935, −10.03151320563821, −9.639726470872890, −8.800289700352483, −8.231635928769068, −7.680271694567016, −7.251313792363088, −6.458672832629183, −6.128199724146251, −5.397983957261981, −5.054867131432142, −4.639677670561611, −3.716068081186972, −3.116830325813911, −2.436116004842388, −1.409104599745157, −0.9194585274592567, 0,
0.9194585274592567, 1.409104599745157, 2.436116004842388, 3.116830325813911, 3.716068081186972, 4.639677670561611, 5.054867131432142, 5.397983957261981, 6.128199724146251, 6.458672832629183, 7.251313792363088, 7.680271694567016, 8.231635928769068, 8.800289700352483, 9.639726470872890, 10.03151320563821, 10.33151086759935, 11.15472713190597, 11.49947368347028, 11.88861666616304, 12.34994123324681, 12.76588524091293, 13.49623325892344, 14.09281108965422, 14.38775632971249