| L(s) = 1 | + 3·3-s − 4·5-s − 10·7-s + 9·9-s + 4·11-s − 26·13-s − 12·15-s + 14·17-s − 8·19-s − 30·21-s − 148·23-s − 109·25-s + 27·27-s − 72·29-s − 18·31-s + 12·33-s + 40·35-s − 262·37-s − 78·39-s − 378·41-s + 432·43-s − 36·45-s − 148·47-s − 243·49-s + 42·51-s − 360·53-s − 16·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.357·5-s − 0.539·7-s + 1/3·9-s + 0.109·11-s − 0.554·13-s − 0.206·15-s + 0.199·17-s − 0.0965·19-s − 0.311·21-s − 1.34·23-s − 0.871·25-s + 0.192·27-s − 0.461·29-s − 0.104·31-s + 0.0633·33-s + 0.193·35-s − 1.16·37-s − 0.320·39-s − 1.43·41-s + 1.53·43-s − 0.119·45-s − 0.459·47-s − 0.708·49-s + 0.115·51-s − 0.933·53-s − 0.0392·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| good | 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + 10 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 T + p^{3} T^{2} \) |
| 13 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 14 T + p^{3} T^{2} \) |
| 19 | \( 1 + 8 T + p^{3} T^{2} \) |
| 23 | \( 1 + 148 T + p^{3} T^{2} \) |
| 29 | \( 1 + 72 T + p^{3} T^{2} \) |
| 31 | \( 1 + 18 T + p^{3} T^{2} \) |
| 37 | \( 1 + 262 T + p^{3} T^{2} \) |
| 41 | \( 1 + 378 T + p^{3} T^{2} \) |
| 43 | \( 1 - 432 T + p^{3} T^{2} \) |
| 47 | \( 1 + 148 T + p^{3} T^{2} \) |
| 53 | \( 1 + 360 T + p^{3} T^{2} \) |
| 59 | \( 1 - 428 T + p^{3} T^{2} \) |
| 61 | \( 1 - 442 T + p^{3} T^{2} \) |
| 67 | \( 1 - 692 T + p^{3} T^{2} \) |
| 71 | \( 1 + 540 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1018 T + p^{3} T^{2} \) |
| 79 | \( 1 + 386 T + p^{3} T^{2} \) |
| 83 | \( 1 + 108 T + p^{3} T^{2} \) |
| 89 | \( 1 + 382 T + p^{3} T^{2} \) |
| 97 | \( 1 - 298 T + p^{3} T^{2} \) |
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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
−10.20809034507253102850493364612, −9.630420353964047726154764252350, −8.555609913996949446313336361067, −7.71221605659445794361028977643, −6.78999694717171069877613859552, −5.60904774918697627530684390452, −4.23158985244375619069545080073, −3.29022808783500250469754186637, −1.94367033478099534047786097643, 0,
1.94367033478099534047786097643, 3.29022808783500250469754186637, 4.23158985244375619069545080073, 5.60904774918697627530684390452, 6.78999694717171069877613859552, 7.71221605659445794361028977643, 8.555609913996949446313336361067, 9.630420353964047726154764252350, 10.20809034507253102850493364612
