| L(s) = 1 | − 2-s − 4-s − 4·5-s + 3·8-s + 4·10-s − 11-s + 2·13-s − 16-s − 6·17-s + 4·19-s + 4·20-s + 22-s − 23-s + 11·25-s − 2·26-s − 2·29-s + 10·31-s − 5·32-s + 6·34-s + 6·37-s − 4·38-s − 12·40-s + 2·41-s − 8·43-s + 44-s + 46-s − 6·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.78·5-s + 1.06·8-s + 1.26·10-s − 0.301·11-s + 0.554·13-s − 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.894·20-s + 0.213·22-s − 0.208·23-s + 11/5·25-s − 0.392·26-s − 0.371·29-s + 1.79·31-s − 0.883·32-s + 1.02·34-s + 0.986·37-s − 0.648·38-s − 1.89·40-s + 0.312·41-s − 1.21·43-s + 0.150·44-s + 0.147·46-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111573 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.64637422117889, −13.43087578692457, −13.04649786224707, −12.19958677059075, −11.91195710702813, −11.42447529613985, −10.84805112689952, −10.64656325521075, −9.903411910761522, −9.342370086506962, −8.889092591722122, −8.374720022529188, −7.988135452103729, −7.669261645832144, −7.090938772891921, −6.523437981414516, −5.898227223195945, −4.930791249710643, −4.544144171668880, −4.316575121588999, −3.444117674207735, −3.156409262807287, −2.196257422606069, −1.278188204021726, −0.6324101852161702, 0,
0.6324101852161702, 1.278188204021726, 2.196257422606069, 3.156409262807287, 3.444117674207735, 4.316575121588999, 4.544144171668880, 4.930791249710643, 5.898227223195945, 6.523437981414516, 7.090938772891921, 7.669261645832144, 7.988135452103729, 8.374720022529188, 8.889092591722122, 9.342370086506962, 9.903411910761522, 10.64656325521075, 10.84805112689952, 11.42447529613985, 11.91195710702813, 12.19958677059075, 13.04649786224707, 13.43087578692457, 13.64637422117889
