| L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s + 7-s + 3·8-s + 9-s + 10-s + 12-s − 2·13-s − 14-s + 15-s − 16-s − 17-s − 18-s + 20-s − 21-s + 4·23-s − 3·24-s + 25-s + 2·26-s − 27-s − 28-s − 6·29-s − 30-s − 5·32-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 + 0.288·12-s − 0.554·13-s − 0.267·14-s + 0.258·15-s − 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.223·20-s − 0.218·21-s + 0.834·23-s − 0.612·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.182·30-s − 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6070715547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6070715547\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.140578695270388175579151998321, −8.684615318586089138051923847400, −7.56027727991688862934645299344, −7.32556938854746287746667675015, −6.08179977882619530602071693949, −5.04102037535368753973952638358, −4.53446601517680007364682783907, −3.47023547889091835658046374013, −1.89996468146154077929099124513, −0.61796632486503105052239820723,
0.61796632486503105052239820723, 1.89996468146154077929099124513, 3.47023547889091835658046374013, 4.53446601517680007364682783907, 5.04102037535368753973952638358, 6.08179977882619530602071693949, 7.32556938854746287746667675015, 7.56027727991688862934645299344, 8.684615318586089138051923847400, 9.140578695270388175579151998321
