| L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s − 7-s + 8-s − 2·9-s − 3·10-s + 12-s − 14-s − 3·15-s + 16-s + 6·17-s − 2·18-s + 4·19-s − 3·20-s − 21-s + 3·23-s + 24-s + 4·25-s − 5·27-s − 28-s + 6·29-s − 3·30-s + 10·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.948·10-s + 0.288·12-s − 0.267·14-s − 0.774·15-s + 1/4·16-s + 1.45·17-s − 0.471·18-s + 0.917·19-s − 0.670·20-s − 0.218·21-s + 0.625·23-s + 0.204·24-s + 4/5·25-s − 0.962·27-s − 0.188·28-s + 1.11·29-s − 0.547·30-s + 1.79·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.549197136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.549197136\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−8.723234695808501521366802911653, −8.116177554606699062383197527473, −7.50882438968180358884564683060, −6.74794137557946929139559683076, −5.71576798840540903108459269111, −4.93980260346123450500517401986, −3.91972626332085563597891162581, −3.26596370742597414894955978659, −2.70130097359635355412487457619, −0.909767728710544922775621814922,
0.909767728710544922775621814922, 2.70130097359635355412487457619, 3.26596370742597414894955978659, 3.91972626332085563597891162581, 4.93980260346123450500517401986, 5.71576798840540903108459269111, 6.74794137557946929139559683076, 7.50882438968180358884564683060, 8.116177554606699062383197527473, 8.723234695808501521366802911653
