| L(s) = 1 | + 3-s + 9-s − 2·11-s − 4·13-s + 17-s − 23-s + 27-s − 2·29-s + 5·31-s − 2·33-s + 4·37-s − 4·39-s − 9·41-s − 4·43-s + 11·47-s + 51-s + 6·59-s − 6·61-s − 69-s − 3·71-s + 6·73-s + 15·79-s + 81-s − 6·83-s − 2·87-s + 9·89-s + 5·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.10·13-s + 0.242·17-s − 0.208·23-s + 0.192·27-s − 0.371·29-s + 0.898·31-s − 0.348·33-s + 0.657·37-s − 0.640·39-s − 1.40·41-s − 0.609·43-s + 1.60·47-s + 0.140·51-s + 0.781·59-s − 0.768·61-s − 0.120·69-s − 0.356·71-s + 0.702·73-s + 1.68·79-s + 1/9·81-s − 0.658·83-s − 0.214·87-s + 0.953·89-s + 0.518·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.74931070641814, −13.88612901919019, −13.75707986638873, −13.12815371379212, −12.54805894316461, −12.13472073865933, −11.65421009027950, −10.97344294648860, −10.28862642004632, −10.06721600990137, −9.443295684854103, −8.963018068915262, −8.260932284081428, −7.889955601838827, −7.363745471401572, −6.826249516892510, −6.204044768955121, −5.428758249890394, −4.995171168293534, −4.368746233975189, −3.713558961724855, −3.008849186040457, −2.471562076215675, −1.910580042943195, −0.9487573095089158, 0,
0.9487573095089158, 1.910580042943195, 2.471562076215675, 3.008849186040457, 3.713558961724855, 4.368746233975189, 4.995171168293534, 5.428758249890394, 6.204044768955121, 6.826249516892510, 7.363745471401572, 7.889955601838827, 8.260932284081428, 8.963018068915262, 9.443295684854103, 10.06721600990137, 10.28862642004632, 10.97344294648860, 11.65421009027950, 12.13472073865933, 12.54805894316461, 13.12815371379212, 13.75707986638873, 13.88612901919019, 14.74931070641814
