L(s) = 1 | − 3·3-s − 2·5-s + 6·9-s + 2·11-s + 13-s + 6·15-s − 2·19-s − 23-s − 25-s − 9·27-s − 3·29-s + 31-s − 6·33-s − 2·37-s − 3·39-s + 41-s + 8·43-s − 12·45-s + 5·47-s − 6·53-s − 4·55-s + 6·57-s − 6·61-s − 2·65-s − 10·67-s + 3·69-s + 7·71-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 0.894·5-s + 2·9-s + 0.603·11-s + 0.277·13-s + 1.54·15-s − 0.458·19-s − 0.208·23-s − 1/5·25-s − 1.73·27-s − 0.557·29-s + 0.179·31-s − 1.04·33-s − 0.328·37-s − 0.480·39-s + 0.156·41-s + 1.21·43-s − 1.78·45-s + 0.729·47-s − 0.824·53-s − 0.539·55-s + 0.794·57-s − 0.768·61-s − 0.248·65-s − 1.22·67-s + 0.361·69-s + 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−7.34474196494962652859719830189, −6.54794417722157833766727440735, −6.04042705154153785335201589228, −5.45317162333408381303036364241, −4.49121613446264497911191953258, −4.19338579186049190745776729474, −3.30712699047776495520446293105, −1.89999359478076841614227986426, −0.901382502735000330137450289202, 0,
0.901382502735000330137450289202, 1.89999359478076841614227986426, 3.30712699047776495520446293105, 4.19338579186049190745776729474, 4.49121613446264497911191953258, 5.45317162333408381303036364241, 6.04042705154153785335201589228, 6.54794417722157833766727440735, 7.34474196494962652859719830189