L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 2·7-s − 8-s + 9-s + 10-s − 12-s + 2·13-s − 2·14-s + 15-s + 16-s − 18-s − 19-s − 20-s − 2·21-s − 8·23-s + 24-s + 25-s − 2·26-s − 27-s + 2·28-s + 2·29-s − 30-s + 8·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.554·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.436·21-s − 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.377·28-s + 0.371·29-s − 0.182·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.63544473406815, −12.80375614840762, −12.22040187135268, −12.11647053554001, −11.50608100673467, −11.01364526543469, −10.76479156367597, −10.24082875192949, −9.616716393543469, −9.324581009491352, −8.417624149749045, −8.251162185765193, −7.802979055974506, −7.336340945721101, −6.553841141539269, −6.246062336175323, −5.807879279621731, −5.005079836651111, −4.592812580290254, −4.002722416606051, −3.478100225453970, −2.591986223685610, −2.088675632975312, −1.324213413884164, −0.8099022724623825, 0,
0.8099022724623825, 1.324213413884164, 2.088675632975312, 2.591986223685610, 3.478100225453970, 4.002722416606051, 4.592812580290254, 5.005079836651111, 5.807879279621731, 6.246062336175323, 6.553841141539269, 7.336340945721101, 7.802979055974506, 8.251162185765193, 8.417624149749045, 9.324581009491352, 9.616716393543469, 10.24082875192949, 10.76479156367597, 11.01364526543469, 11.50608100673467, 12.11647053554001, 12.22040187135268, 12.80375614840762, 13.63544473406815