| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 3·7-s − 8-s + 9-s − 10-s − 12-s + 13-s + 3·14-s − 15-s + 16-s + 2·17-s − 18-s + 19-s + 20-s + 3·21-s − 4·23-s + 24-s + 25-s − 26-s − 27-s − 3·28-s − 6·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.277·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.654·21-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.566·28-s − 1.11·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.95366951100415, −14.36401078180372, −13.74433951671531, −13.11734568090255, −12.87656186993843, −12.11089921464498, −11.79517337026505, −11.15913654680953, −10.53248698255908, −10.13331126379190, −9.669234430156163, −9.251798857574951, −8.676295783709455, −7.882765792365745, −7.474240905489755, −6.770177879599327, −6.234460294094185, −5.938635144931690, −5.290896073428095, −4.517854194651401, −3.645145315803215, −3.210506914458489, −2.346706770247336, −1.649998331641282, −0.8052522169178600, 0,
0.8052522169178600, 1.649998331641282, 2.346706770247336, 3.210506914458489, 3.645145315803215, 4.517854194651401, 5.290896073428095, 5.938635144931690, 6.234460294094185, 6.770177879599327, 7.474240905489755, 7.882765792365745, 8.676295783709455, 9.251798857574951, 9.669234430156163, 10.13331126379190, 10.53248698255908, 11.15913654680953, 11.79517337026505, 12.11089921464498, 12.87656186993843, 13.11734568090255, 13.74433951671531, 14.36401078180372, 14.95366951100415
