| L(s) = 1 | + 3-s − 2·5-s + 9-s + 13-s − 2·15-s − 17-s + 2·19-s − 25-s + 27-s − 2·29-s + 8·31-s + 4·37-s + 39-s − 10·41-s − 2·45-s − 7·49-s − 51-s + 6·53-s + 2·57-s − 14·61-s − 2·65-s − 6·67-s + 6·71-s − 4·73-s − 75-s + 8·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.277·13-s − 0.516·15-s − 0.242·17-s + 0.458·19-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.657·37-s + 0.160·39-s − 1.56·41-s − 0.298·45-s − 49-s − 0.140·51-s + 0.824·53-s + 0.264·57-s − 1.79·61-s − 0.248·65-s − 0.733·67-s + 0.712·71-s − 0.468·73-s − 0.115·75-s + 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.88440717315032, −12.32031663970101, −11.89649074813707, −11.60668360373167, −11.05409497980581, −10.64000386329240, −9.971138651902090, −9.732863222110321, −9.142170849995676, −8.554931198149262, −8.331383346987121, −7.713146670441039, −7.526888342107864, −6.806850020798795, −6.459856562899966, −5.844369741526736, −5.233654591682197, −4.636688415225073, −4.237833877879181, −3.757665586887557, −3.093734599666578, −2.896965627817661, −2.009578493822262, −1.492050337943157, −0.7449930965738251, 0,
0.7449930965738251, 1.492050337943157, 2.009578493822262, 2.896965627817661, 3.093734599666578, 3.757665586887557, 4.237833877879181, 4.636688415225073, 5.233654591682197, 5.844369741526736, 6.459856562899966, 6.806850020798795, 7.526888342107864, 7.713146670441039, 8.331383346987121, 8.554931198149262, 9.142170849995676, 9.732863222110321, 9.971138651902090, 10.64000386329240, 11.05409497980581, 11.60668360373167, 11.89649074813707, 12.32031663970101, 12.88440717315032
