| L(s) = 1 | + 3-s + 9-s + 4·11-s + 6·13-s − 6·17-s − 4·19-s + 8·23-s + 27-s + 10·29-s − 4·31-s + 4·33-s + 6·37-s + 6·39-s − 6·41-s − 4·43-s − 12·47-s − 6·51-s − 6·53-s − 4·57-s − 4·59-s + 2·61-s − 4·67-s + 8·69-s − 2·73-s − 8·79-s + 81-s − 12·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 1.45·17-s − 0.917·19-s + 1.66·23-s + 0.192·27-s + 1.85·29-s − 0.718·31-s + 0.696·33-s + 0.986·37-s + 0.960·39-s − 0.937·41-s − 0.609·43-s − 1.75·47-s − 0.840·51-s − 0.824·53-s − 0.529·57-s − 0.520·59-s + 0.256·61-s − 0.488·67-s + 0.963·69-s − 0.234·73-s − 0.900·79-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.71460448764819, −13.39554041510432, −12.90364750230850, −12.64256778812786, −11.70041417458116, −11.39782997920925, −10.98513031925624, −10.50166521354432, −9.859793212457494, −9.259272736491048, −8.828051727402283, −8.487958340107935, −8.225763761912122, −7.237239737243345, −6.752600801953442, −6.413529100293308, −6.059255319420556, −5.054865635305909, −4.495455591110282, −4.172869382321188, −3.391737782089678, −3.061675146375939, −2.251890760882906, −1.447328366725680, −1.173548139034654, 0,
1.173548139034654, 1.447328366725680, 2.251890760882906, 3.061675146375939, 3.391737782089678, 4.172869382321188, 4.495455591110282, 5.054865635305909, 6.059255319420556, 6.413529100293308, 6.752600801953442, 7.237239737243345, 8.225763761912122, 8.487958340107935, 8.828051727402283, 9.259272736491048, 9.859793212457494, 10.50166521354432, 10.98513031925624, 11.39782997920925, 11.70041417458116, 12.64256778812786, 12.90364750230850, 13.39554041510432, 13.71460448764819
