| L(s) = 1 | + 2-s + 4-s − 2·5-s − 7-s + 8-s − 2·10-s + 11-s − 4·13-s − 14-s + 16-s − 2·17-s − 6·19-s − 2·20-s + 22-s + 2·23-s − 25-s − 4·26-s − 28-s + 2·29-s − 8·31-s + 32-s − 2·34-s + 2·35-s − 2·37-s − 6·38-s − 2·40-s + 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s + 0.353·8-s − 0.632·10-s + 0.301·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 1.37·19-s − 0.447·20-s + 0.213·22-s + 0.417·23-s − 1/5·25-s − 0.784·26-s − 0.188·28-s + 0.371·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s + 0.338·35-s − 0.328·37-s − 0.973·38-s − 0.316·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−9.134494299607741468141713783567, −8.272162289910099551122345576861, −7.34142071798424504356409604185, −6.78155434392206371572794646550, −5.81531940011466945768947277432, −4.73109850526366519062683146302, −4.09136916114338661687753966711, −3.14508766922258374979544108268, −2.02258435897521586180281099584, 0,
2.02258435897521586180281099584, 3.14508766922258374979544108268, 4.09136916114338661687753966711, 4.73109850526366519062683146302, 5.81531940011466945768947277432, 6.78155434392206371572794646550, 7.34142071798424504356409604185, 8.272162289910099551122345576861, 9.134494299607741468141713783567
