| L(s) = 1 | + 4·5-s + 2·7-s + 11-s − 13-s + 17-s + 4·19-s − 6·23-s + 11·25-s − 10·31-s + 8·35-s + 12·37-s − 2·41-s + 12·43-s − 4·47-s − 3·49-s − 6·53-s + 4·55-s + 4·59-s − 8·61-s − 4·65-s − 12·67-s + 2·71-s − 10·73-s + 2·77-s + 2·79-s + 4·83-s + 4·85-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.755·7-s + 0.301·11-s − 0.277·13-s + 0.242·17-s + 0.917·19-s − 1.25·23-s + 11/5·25-s − 1.79·31-s + 1.35·35-s + 1.97·37-s − 0.312·41-s + 1.82·43-s − 0.583·47-s − 3/7·49-s − 0.824·53-s + 0.539·55-s + 0.520·59-s − 1.02·61-s − 0.496·65-s − 1.46·67-s + 0.237·71-s − 1.17·73-s + 0.227·77-s + 0.225·79-s + 0.439·83-s + 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−12.87437478940072, −12.43333035609180, −11.80338025291261, −11.39819277783744, −10.95557642837700, −10.38628063039044, −10.06795957999905, −9.538486806569173, −9.170420660552136, −8.971086260344035, −8.107167658794294, −7.654662866077663, −7.396522339712445, −6.591855673313159, −6.128762516210416, −5.812793853127967, −5.370631407787093, −4.874528430140602, −4.358712379569254, −3.743108163677239, −3.002360988042711, −2.503853381101204, −1.974901235666118, −1.460315506079767, −1.081683600932767, 0,
1.081683600932767, 1.460315506079767, 1.974901235666118, 2.503853381101204, 3.002360988042711, 3.743108163677239, 4.358712379569254, 4.874528430140602, 5.370631407787093, 5.812793853127967, 6.128762516210416, 6.591855673313159, 7.396522339712445, 7.654662866077663, 8.107167658794294, 8.971086260344035, 9.170420660552136, 9.538486806569173, 10.06795957999905, 10.38628063039044, 10.95557642837700, 11.39819277783744, 11.80338025291261, 12.43333035609180, 12.87437478940072
