| L(s) = 1 | − 3-s − 5-s + 9-s − 4·11-s − 2·13-s + 15-s − 4·19-s + 25-s − 27-s + 2·29-s + 4·33-s + 10·37-s + 2·39-s − 10·41-s − 4·43-s − 45-s − 8·47-s − 7·49-s − 10·53-s + 4·55-s + 4·57-s + 4·59-s + 2·61-s + 2·65-s − 12·67-s − 8·71-s − 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.258·15-s − 0.917·19-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.696·33-s + 1.64·37-s + 0.320·39-s − 1.56·41-s − 0.609·43-s − 0.149·45-s − 1.16·47-s − 49-s − 1.37·53-s + 0.539·55-s + 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.248·65-s − 1.46·67-s − 0.949·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.66308059562493, −14.33626527833895, −13.34144263481564, −13.09853606534291, −12.78307093043810, −12.01558445463326, −11.70346881065831, −11.12986969068413, −10.68299070318667, −10.05809451631012, −9.864619370793713, −9.037807988416762, −8.343703824484660, −8.019530889524605, −7.477414175588173, −6.842926109203982, −6.369678398070988, −5.753132270001528, −5.118776607415726, −4.625130049876126, −4.251975109039251, −3.235393930299933, −2.847901830115221, −2.010664178150182, −1.283485399829065, 0, 0,
1.283485399829065, 2.010664178150182, 2.847901830115221, 3.235393930299933, 4.251975109039251, 4.625130049876126, 5.118776607415726, 5.753132270001528, 6.369678398070988, 6.842926109203982, 7.477414175588173, 8.019530889524605, 8.343703824484660, 9.037807988416762, 9.864619370793713, 10.05809451631012, 10.68299070318667, 11.12986969068413, 11.70346881065831, 12.01558445463326, 12.78307093043810, 13.09853606534291, 13.34144263481564, 14.33626527833895, 14.66308059562493
