| L(s) = 1 | − 24·7-s + 44·11-s − 22·13-s + 50·17-s + 44·19-s − 56·23-s − 198·29-s − 160·31-s + 162·37-s + 198·41-s − 52·43-s + 528·47-s + 233·49-s − 242·53-s + 668·59-s + 550·61-s − 188·67-s − 728·71-s − 154·73-s − 1.05e3·77-s − 656·79-s + 236·83-s − 714·89-s + 528·91-s + 478·97-s − 1.56e3·101-s + 968·103-s + ⋯ |
| L(s) = 1 | − 1.29·7-s + 1.20·11-s − 0.469·13-s + 0.713·17-s + 0.531·19-s − 0.507·23-s − 1.26·29-s − 0.926·31-s + 0.719·37-s + 0.754·41-s − 0.184·43-s + 1.63·47-s + 0.679·49-s − 0.627·53-s + 1.47·59-s + 1.15·61-s − 0.342·67-s − 1.21·71-s − 0.246·73-s − 1.56·77-s − 0.934·79-s + 0.312·83-s − 0.850·89-s + 0.608·91-s + 0.500·97-s − 1.54·101-s + 0.926·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 24 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 - 50 T + p^{3} T^{2} \) |
| 19 | \( 1 - 44 T + p^{3} T^{2} \) |
| 23 | \( 1 + 56 T + p^{3} T^{2} \) |
| 29 | \( 1 + 198 T + p^{3} T^{2} \) |
| 31 | \( 1 + 160 T + p^{3} T^{2} \) |
| 37 | \( 1 - 162 T + p^{3} T^{2} \) |
| 41 | \( 1 - 198 T + p^{3} T^{2} \) |
| 43 | \( 1 + 52 T + p^{3} T^{2} \) |
| 47 | \( 1 - 528 T + p^{3} T^{2} \) |
| 53 | \( 1 + 242 T + p^{3} T^{2} \) |
| 59 | \( 1 - 668 T + p^{3} T^{2} \) |
| 61 | \( 1 - 550 T + p^{3} T^{2} \) |
| 67 | \( 1 + 188 T + p^{3} T^{2} \) |
| 71 | \( 1 + 728 T + p^{3} T^{2} \) |
| 73 | \( 1 + 154 T + p^{3} T^{2} \) |
| 79 | \( 1 + 656 T + p^{3} T^{2} \) |
| 83 | \( 1 - 236 T + p^{3} T^{2} \) |
| 89 | \( 1 + 714 T + p^{3} T^{2} \) |
| 97 | \( 1 - 478 T + p^{3} T^{2} \) |
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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
−8.687147679266628718588888118938, −7.52897034336328746954900834937, −6.99768584104102333715941849518, −6.07246550045469698457847110707, −5.48923955664485230872755505642, −4.11217773970782304976909373890, −3.54470403181847861116617730865, −2.50824677694422533718193916738, −1.20808733640550797019789221265, 0,
1.20808733640550797019789221265, 2.50824677694422533718193916738, 3.54470403181847861116617730865, 4.11217773970782304976909373890, 5.48923955664485230872755505642, 6.07246550045469698457847110707, 6.99768584104102333715941849518, 7.52897034336328746954900834937, 8.687147679266628718588888118938
