| L(s) = 1 | − 5-s + 3·7-s + 4·11-s − 3·13-s − 7·19-s − 4·23-s + 25-s − 29-s − 3·31-s − 3·35-s + 12·37-s − 9·41-s − 43-s + 6·47-s + 2·49-s + 6·53-s − 4·55-s − 4·59-s + 3·61-s + 3·65-s − 67-s − 10·71-s + 11·73-s + 12·77-s − 13·79-s + 16·89-s − 9·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s + 1.20·11-s − 0.832·13-s − 1.60·19-s − 0.834·23-s + 1/5·25-s − 0.185·29-s − 0.538·31-s − 0.507·35-s + 1.97·37-s − 1.40·41-s − 0.152·43-s + 0.875·47-s + 2/7·49-s + 0.824·53-s − 0.539·55-s − 0.520·59-s + 0.384·61-s + 0.372·65-s − 0.122·67-s − 1.18·71-s + 1.28·73-s + 1.36·77-s − 1.46·79-s + 1.69·89-s − 0.943·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−15.06475787597162, −14.90006404923765, −14.39423930518896, −14.01456072721521, −13.14535006173701, −12.71074774116916, −12.02461821045653, −11.60694256911744, −11.31728869311524, −10.51584404591205, −10.12190818696876, −9.329914992865825, −8.762951581352690, −8.368757247829452, −7.625303443378405, −7.327577037200786, −6.414750658074691, −6.087344665205107, −5.137061107857299, −4.564689886489495, −4.125076778631514, −3.522009541229120, −2.398380641937528, −1.934118179096180, −1.072658516918371, 0,
1.072658516918371, 1.934118179096180, 2.398380641937528, 3.522009541229120, 4.125076778631514, 4.564689886489495, 5.137061107857299, 6.087344665205107, 6.414750658074691, 7.327577037200786, 7.625303443378405, 8.368757247829452, 8.762951581352690, 9.329914992865825, 10.12190818696876, 10.51584404591205, 11.31728869311524, 11.60694256911744, 12.02461821045653, 12.71074774116916, 13.14535006173701, 14.01456072721521, 14.39423930518896, 14.90006404923765, 15.06475787597162
