| L(s) = 1 | − 3-s + 5-s + 3·7-s − 2·9-s + 4·11-s + 7·13-s − 15-s − 4·17-s − 6·19-s − 3·21-s − 6·23-s − 4·25-s + 5·27-s − 4·29-s − 8·31-s − 4·33-s + 3·35-s − 10·37-s − 7·39-s − 8·41-s − 8·43-s − 2·45-s + 3·47-s + 2·49-s + 4·51-s − 2·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.13·7-s − 2/3·9-s + 1.20·11-s + 1.94·13-s − 0.258·15-s − 0.970·17-s − 1.37·19-s − 0.654·21-s − 1.25·23-s − 4/5·25-s + 0.962·27-s − 0.742·29-s − 1.43·31-s − 0.696·33-s + 0.507·35-s − 1.64·37-s − 1.12·39-s − 1.24·41-s − 1.21·43-s − 0.298·45-s + 0.437·47-s + 2/7·49-s + 0.560·51-s − 0.274·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−8.126957497976974844919426109083, −6.90476391393474938494772777925, −6.29740142055844636244112916420, −5.82520855040424885855116690941, −5.05765838785199927476622211550, −4.06142561404756926074204719353, −3.59967036868775553742439514434, −1.89745981147512300655427994206, −1.64842617233660497870298811046, 0,
1.64842617233660497870298811046, 1.89745981147512300655427994206, 3.59967036868775553742439514434, 4.06142561404756926074204719353, 5.05765838785199927476622211550, 5.82520855040424885855116690941, 6.29740142055844636244112916420, 6.90476391393474938494772777925, 8.126957497976974844919426109083
