| L(s) = 1 | − 2-s − 4-s − 2·5-s + 3·8-s + 2·10-s + 4·11-s + 2·13-s − 16-s − 2·17-s − 4·19-s + 2·20-s − 4·22-s − 25-s − 2·26-s − 8·31-s − 5·32-s + 2·34-s + 10·37-s + 4·38-s − 6·40-s + 6·41-s − 12·43-s − 4·44-s + 8·47-s + 50-s − 2·52-s − 6·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.06·8-s + 0.632·10-s + 1.20·11-s + 0.554·13-s − 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.447·20-s − 0.852·22-s − 1/5·25-s − 0.392·26-s − 1.43·31-s − 0.883·32-s + 0.342·34-s + 1.64·37-s + 0.648·38-s − 0.948·40-s + 0.937·41-s − 1.82·43-s − 0.603·44-s + 1.16·47-s + 0.141·50-s − 0.277·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.73037518459072, −12.22370890551101, −11.72969040885301, −11.21927570285727, −10.97896919357401, −10.52750026520540, −9.881022216010612, −9.414011209657253, −9.075484450483934, −8.691608666305361, −8.211296235196804, −7.799971317489451, −7.382068323521286, −6.811391165529448, −6.333217019497670, −5.875208005109747, −5.152942341816017, −4.584522250060769, −4.099572447716293, −3.859740619875371, −3.382249976341168, −2.463430574010635, −1.826822451676147, −1.275320925007959, −0.6207580589691913, 0,
0.6207580589691913, 1.275320925007959, 1.826822451676147, 2.463430574010635, 3.382249976341168, 3.859740619875371, 4.099572447716293, 4.584522250060769, 5.152942341816017, 5.875208005109747, 6.333217019497670, 6.811391165529448, 7.382068323521286, 7.799971317489451, 8.211296235196804, 8.691608666305361, 9.075484450483934, 9.414011209657253, 9.881022216010612, 10.52750026520540, 10.97896919357401, 11.21927570285727, 11.72969040885301, 12.22370890551101, 12.73037518459072
