| L(s) = 1 | − 3-s + 9-s + 4·11-s − 4·13-s − 4·19-s − 27-s + 2·29-s − 8·31-s − 4·33-s + 6·37-s + 4·39-s + 4·43-s − 8·47-s + 10·53-s + 4·57-s − 4·59-s − 4·61-s + 4·67-s − 8·71-s + 16·73-s + 8·79-s + 81-s − 12·83-s − 2·87-s + 8·89-s + 8·93-s − 8·97-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s − 1.10·13-s − 0.917·19-s − 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.696·33-s + 0.986·37-s + 0.640·39-s + 0.609·43-s − 1.16·47-s + 1.37·53-s + 0.529·57-s − 0.520·59-s − 0.512·61-s + 0.488·67-s − 0.949·71-s + 1.87·73-s + 0.900·79-s + 1/9·81-s − 1.31·83-s − 0.214·87-s + 0.847·89-s + 0.829·93-s − 0.812·97-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 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.59167220786212, −14.23919881743565, −13.54663651784923, −12.89611499596319, −12.55710876248192, −12.00998202741388, −11.64112869096971, −10.97200191694162, −10.67945802081798, −9.873989537782786, −9.548914035264595, −9.003806957066394, −8.429161822161964, −7.687503049200187, −7.246535153929835, −6.579038788367849, −6.300339543197634, −5.524067440862365, −5.034730579785750, −4.289971812943860, −3.987949055715287, −3.137132271465042, −2.315226769779824, −1.721140614523720, −0.8626828662143809, 0,
0.8626828662143809, 1.721140614523720, 2.315226769779824, 3.137132271465042, 3.987949055715287, 4.289971812943860, 5.034730579785750, 5.524067440862365, 6.300339543197634, 6.579038788367849, 7.246535153929835, 7.687503049200187, 8.429161822161964, 9.003806957066394, 9.548914035264595, 9.873989537782786, 10.67945802081798, 10.97200191694162, 11.64112869096971, 12.00998202741388, 12.55710876248192, 12.89611499596319, 13.54663651784923, 14.23919881743565, 14.59167220786212
