| L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s + 7-s + 3·8-s + 9-s + 10-s + 11-s + 12-s − 2·13-s − 14-s + 15-s − 16-s − 18-s + 4·19-s + 20-s − 21-s − 22-s − 8·23-s − 3·24-s + 25-s + 2·26-s − 27-s − 28-s + 2·29-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 + 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.554·13-s − 0.267·14-s + 0.258·15-s − 1/4·16-s − 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.218·21-s − 0.213·22-s − 1.66·23-s − 0.612·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 333795 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333795 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−12.55089282017174, −12.20003912879610, −11.98461747642579, −11.45374210157406, −10.98143554000883, −10.34734417074385, −10.05002521177508, −9.843212800962424, −9.100852968589883, −8.642585610337167, −8.357714329595077, −7.635432442087424, −7.473844259401952, −7.003020727854822, −6.155425984058574, −5.965630769314753, −5.049152317510339, −4.887105281264291, −4.372686899514169, −3.791374358639089, −3.340099342602315, −2.457540047268593, −1.834251771181093, −1.217239100542020, −0.6414975056374544, 0,
0.6414975056374544, 1.217239100542020, 1.834251771181093, 2.457540047268593, 3.340099342602315, 3.791374358639089, 4.372686899514169, 4.887105281264291, 5.049152317510339, 5.965630769314753, 6.155425984058574, 7.003020727854822, 7.473844259401952, 7.635432442087424, 8.357714329595077, 8.642585610337167, 9.100852968589883, 9.843212800962424, 10.05002521177508, 10.34734417074385, 10.98143554000883, 11.45374210157406, 11.98461747642579, 12.20003912879610, 12.55089282017174
