| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s + 4·7-s − 8-s + 9-s − 2·10-s − 4·11-s − 12-s + 13-s − 4·14-s − 2·15-s + 16-s + 2·17-s − 18-s − 8·19-s + 2·20-s − 4·21-s + 4·22-s + 24-s − 25-s − 26-s − 27-s + 4·28-s + 6·29-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 − 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s − 0.288·12-s + 0.277·13-s − 1.06·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.83·19-s + 0.447·20-s − 0.872·21-s + 0.852·22-s + 0.204·24-s − 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.755·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7252179630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7252179630\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.56548660784257462371286700962, −13.34702172357780953528143324388, −12.08425672990514369865643110699, −10.80469174604554316475352751747, −10.33346044923143313228856689357, −8.724491555695013698121297530874, −7.70223650354535578507572334847, −6.10664918750757402239876678098, −4.91686329817785172521042266230, −1.96145678743744747791090034638,
1.96145678743744747791090034638, 4.91686329817785172521042266230, 6.10664918750757402239876678098, 7.70223650354535578507572334847, 8.724491555695013698121297530874, 10.33346044923143313228856689357, 10.80469174604554316475352751747, 12.08425672990514369865643110699, 13.34702172357780953528143324388, 14.56548660784257462371286700962
