L(s) = 1 | + 2-s − 4-s − 5-s − 7-s − 3·8-s − 10-s − 11-s − 2·13-s − 14-s − 16-s + 6·17-s − 4·19-s + 20-s − 22-s − 8·23-s + 25-s − 2·26-s + 28-s − 6·29-s + 8·31-s + 5·32-s + 6·34-s + 35-s + 6·37-s − 4·38-s + 3·40-s + 6·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.377·7-s − 1.06·8-s − 0.316·10-s − 0.301·11-s − 0.554·13-s − 0.267·14-s − 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.223·20-s − 0.213·22-s − 1.66·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.883·32-s + 1.02·34-s + 0.169·35-s + 0.986·37-s − 0.648·38-s + 0.474·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.366309474\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.366309474\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.431837610884012050796498968048, −7.923081692156930228024485852131, −7.10713035226523842129919724338, −5.96287781432669962376720723213, −5.69649321420583321435891782414, −4.56896563458714726523440348234, −4.05616540825443788590460802815, −3.23319758564095090995047738498, −2.31988759231103471908330769114, −0.60189770272591017492392273116,
0.60189770272591017492392273116, 2.31988759231103471908330769114, 3.23319758564095090995047738498, 4.05616540825443788590460802815, 4.56896563458714726523440348234, 5.69649321420583321435891782414, 5.96287781432669962376720723213, 7.10713035226523842129919724338, 7.923081692156930228024485852131, 8.431837610884012050796498968048