L(s) = 1 | − 2-s + 2·3-s + 4-s + 5-s − 2·6-s − 8-s + 9-s − 10-s − 11-s + 2·12-s − 2·13-s + 2·15-s + 16-s + 6·17-s − 18-s − 2·19-s + 20-s + 22-s − 6·23-s − 2·24-s + 25-s + 2·26-s − 4·27-s − 2·30-s − 8·31-s − 32-s − 2·33-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.577·12-s − 0.554·13-s + 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.213·22-s − 1.25·23-s − 0.408·24-s + 1/5·25-s + 0.392·26-s − 0.769·27-s − 0.365·30-s − 1.43·31-s − 0.176·32-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.100287354722351740118396870835, −7.35539584869822957604594549716, −6.60917017530382664296252335156, −5.67726847630877229411662708881, −5.03214215361775283568750803279, −3.67841618616134056250085124940, −3.20161008208660542164368410867, −2.18066836222259146215022225113, −1.65324664936885486756008330583, 0,
1.65324664936885486756008330583, 2.18066836222259146215022225113, 3.20161008208660542164368410867, 3.67841618616134056250085124940, 5.03214215361775283568750803279, 5.67726847630877229411662708881, 6.60917017530382664296252335156, 7.35539584869822957604594549716, 8.100287354722351740118396870835