| L(s) = 1 | − 2-s + 3-s − 4-s − 2·5-s − 6-s + 4·7-s + 3·8-s + 9-s + 2·10-s + 4·11-s − 12-s − 2·13-s − 4·14-s − 2·15-s − 16-s − 2·17-s − 18-s − 4·19-s + 2·20-s + 4·21-s − 4·22-s + 8·23-s + 3·24-s − 25-s + 2·26-s + 27-s − 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s + 1.51·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s − 1.06·14-s − 0.516·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.872·21-s − 0.852·22-s + 1.66·23-s + 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1293 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1293 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.276662627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.276662627\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.338795935601519861172428876472, −8.712904638552331721593014541932, −8.278576428664202632695394290679, −7.48274748658914157519744729454, −6.80705437421550722495951834705, −5.09681695092309864350554346789, −4.44035962626912379708693878258, −3.75346746879010338129919392826, −2.11343426103611876230432026759, −0.963958871058833045283681000382,
0.963958871058833045283681000382, 2.11343426103611876230432026759, 3.75346746879010338129919392826, 4.44035962626912379708693878258, 5.09681695092309864350554346789, 6.80705437421550722495951834705, 7.48274748658914157519744729454, 8.278576428664202632695394290679, 8.712904638552331721593014541932, 9.338795935601519861172428876472
