| L(s) = 1 | − 3-s + 2·7-s + 9-s − 2·11-s + 2·13-s + 6·17-s − 8·19-s − 2·21-s + 4·23-s − 27-s + 8·29-s + 2·33-s − 10·37-s − 2·39-s + 2·41-s + 12·43-s − 3·49-s − 6·51-s + 10·53-s + 8·57-s + 6·59-s + 2·61-s + 2·63-s + 8·67-s − 4·69-s + 4·71-s + 4·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 1.45·17-s − 1.83·19-s − 0.436·21-s + 0.834·23-s − 0.192·27-s + 1.48·29-s + 0.348·33-s − 1.64·37-s − 0.320·39-s + 0.312·41-s + 1.82·43-s − 3/7·49-s − 0.840·51-s + 1.37·53-s + 1.05·57-s + 0.781·59-s + 0.256·61-s + 0.251·63-s + 0.977·67-s − 0.481·69-s + 0.474·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.473768735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.473768735\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.984594639206834713712861443795, −8.741609197992580996424513741397, −8.195538584985523867923416005559, −7.25884935363543930478100731488, −6.34972018901595726678770534815, −5.45948728133820080364901262125, −4.74224310883135659111871048891, −3.69730656029755600020321293964, −2.33249777746263863841083790985, −0.976070537187603548317524608469,
0.976070537187603548317524608469, 2.33249777746263863841083790985, 3.69730656029755600020321293964, 4.74224310883135659111871048891, 5.45948728133820080364901262125, 6.34972018901595726678770534815, 7.25884935363543930478100731488, 8.195538584985523867923416005559, 8.741609197992580996424513741397, 9.984594639206834713712861443795
