| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s − 2·7-s + 3·8-s + 9-s + 11-s + 12-s + 2·13-s + 2·14-s − 16-s − 18-s + 2·19-s + 2·21-s − 22-s + 23-s − 3·24-s − 5·25-s − 2·26-s − 27-s + 2·28-s − 10·29-s + 4·31-s − 5·32-s − 33-s − 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s − 0.755·7-s + 1.06·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 0.554·13-s + 0.534·14-s − 1/4·16-s − 0.235·18-s + 0.458·19-s + 0.436·21-s − 0.213·22-s + 0.208·23-s − 0.612·24-s − 25-s − 0.392·26-s − 0.192·27-s + 0.377·28-s − 1.85·29-s + 0.718·31-s − 0.883·32-s − 0.174·33-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 759 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 759 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−9.606638633025363247063793989859, −9.418353076850431620460572614619, −8.238566263836883333952812885528, −7.43694775883103025425127148067, −6.40919369488320516035346481826, −5.53728164137839547865498830500, −4.40674000762659818554698704055, −3.41882867041015438831449349930, −1.51326525110059255909766957712, 0,
1.51326525110059255909766957712, 3.41882867041015438831449349930, 4.40674000762659818554698704055, 5.53728164137839547865498830500, 6.40919369488320516035346481826, 7.43694775883103025425127148067, 8.238566263836883333952812885528, 9.418353076850431620460572614619, 9.606638633025363247063793989859
