| L(s) = 1 | + 3-s − 2·4-s + 3·5-s + 7-s − 2·9-s − 11-s − 2·12-s − 4·13-s + 3·15-s + 4·16-s − 6·17-s + 2·19-s − 6·20-s + 21-s + 3·23-s + 4·25-s − 5·27-s − 2·28-s − 6·29-s + 5·31-s − 33-s + 3·35-s + 4·36-s + 11·37-s − 4·39-s + 6·41-s + 8·43-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1.34·5-s + 0.377·7-s − 2/3·9-s − 0.301·11-s − 0.577·12-s − 1.10·13-s + 0.774·15-s + 16-s − 1.45·17-s + 0.458·19-s − 1.34·20-s + 0.218·21-s + 0.625·23-s + 4/5·25-s − 0.962·27-s − 0.377·28-s − 1.11·29-s + 0.898·31-s − 0.174·33-s + 0.507·35-s + 2/3·36-s + 1.80·37-s − 0.640·39-s + 0.937·41-s + 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.032606710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.032606710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.24843777324878748577425819366, −13.60450574831141336956421481732, −12.75123359214433885650815349898, −11.03928612573383528826645274943, −9.594904136117515823873522897188, −9.121939609810073781754575230776, −7.79543673470654027074280114997, −5.89237731397243587720148634695, −4.69940224529243356978622440045, −2.52489627324288051568453078479,
2.52489627324288051568453078479, 4.69940224529243356978622440045, 5.89237731397243587720148634695, 7.79543673470654027074280114997, 9.121939609810073781754575230776, 9.594904136117515823873522897188, 11.03928612573383528826645274943, 12.75123359214433885650815349898, 13.60450574831141336956421481732, 14.24843777324878748577425819366
