L(s) = 1 | − 2·5-s − 2·7-s + 4·11-s − 2·13-s + 4·17-s − 4·19-s + 8·23-s − 25-s − 6·29-s + 6·31-s + 4·35-s + 2·37-s + 12·41-s + 12·43-s + 8·47-s − 3·49-s − 6·53-s − 8·55-s + 8·59-s + 10·61-s + 4·65-s − 8·67-s + 2·73-s − 8·77-s + 14·79-s + 12·83-s − 8·85-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.755·7-s + 1.20·11-s − 0.554·13-s + 0.970·17-s − 0.917·19-s + 1.66·23-s − 1/5·25-s − 1.11·29-s + 1.07·31-s + 0.676·35-s + 0.328·37-s + 1.87·41-s + 1.82·43-s + 1.16·47-s − 3/7·49-s − 0.824·53-s − 1.07·55-s + 1.04·59-s + 1.28·61-s + 0.496·65-s − 0.977·67-s + 0.234·73-s − 0.911·77-s + 1.57·79-s + 1.31·83-s − 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.272871298\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.272871298\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−9.534316576287575327840200190345, −9.197093689735864861931474284301, −8.067304075703030681426439396513, −7.31723999205560521648363711728, −6.56096797266048353283337725893, −5.63503930527170379865234790447, −4.35630788801106525176387816815, −3.71624445910859273703515406596, −2.63082001424136503208341482319, −0.861313696819797789656347648853,
0.861313696819797789656347648853, 2.63082001424136503208341482319, 3.71624445910859273703515406596, 4.35630788801106525176387816815, 5.63503930527170379865234790447, 6.56096797266048353283337725893, 7.31723999205560521648363711728, 8.067304075703030681426439396513, 9.197093689735864861931474284301, 9.534316576287575327840200190345