| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s + 11-s + 12-s − 13-s − 14-s + 15-s + 16-s + 17-s − 18-s − 3·19-s + 20-s + 21-s − 22-s + 23-s − 24-s − 4·25-s + 26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.688·19-s + 0.223·20-s + 0.218·21-s − 0.213·22-s + 0.208·23-s − 0.204·24-s − 4/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−15.75825328804800, −15.16409365362077, −14.85333560058016, −14.09788390896473, −13.82722369257860, −13.05189872701028, −12.55435052481167, −11.86191535057258, −11.41027412480882, −10.65696884731005, −10.22873142729119, −9.534967254792638, −9.284045442949570, −8.502890704242733, −7.989134050312461, −7.639522693542746, −6.675215055482743, −6.375264255581229, −5.574835402427441, −4.738779031696154, −4.178454643098023, −3.202147519513063, −2.639147498458061, −1.798834825676004, −1.275890281550142, 0,
1.275890281550142, 1.798834825676004, 2.639147498458061, 3.202147519513063, 4.178454643098023, 4.738779031696154, 5.574835402427441, 6.375264255581229, 6.675215055482743, 7.639522693542746, 7.989134050312461, 8.502890704242733, 9.284045442949570, 9.534967254792638, 10.22873142729119, 10.65696884731005, 11.41027412480882, 11.86191535057258, 12.55435052481167, 13.05189872701028, 13.82722369257860, 14.09788390896473, 14.85333560058016, 15.16409365362077, 15.75825328804800
