| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 4·5-s + 2·6-s + 7-s + 9-s − 8·10-s − 2·12-s − 2·13-s − 2·14-s − 4·15-s − 4·16-s + 4·17-s − 2·18-s − 3·19-s + 8·20-s − 21-s + 2·23-s + 11·25-s + 4·26-s − 27-s + 2·28-s + 6·29-s + 8·30-s − 5·31-s + 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 1.78·5-s + 0.816·6-s + 0.377·7-s + 1/3·9-s − 2.52·10-s − 0.577·12-s − 0.554·13-s − 0.534·14-s − 1.03·15-s − 16-s + 0.970·17-s − 0.471·18-s − 0.688·19-s + 1.78·20-s − 0.218·21-s + 0.417·23-s + 11/5·25-s + 0.784·26-s − 0.192·27-s + 0.377·28-s + 1.11·29-s + 1.46·30-s − 0.898·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7861735760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7861735760\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−10.86340229289232847766758241404, −10.38252022530035955702643197808, −9.598150501746471856910168767253, −8.972019134073409502580254099142, −7.79298545455764963624452711488, −6.75395532990341575165448572479, −5.80045457911231193627474837008, −4.79085093103530693850477639954, −2.38116356225299620011633985249, −1.22560267668749840643223616905,
1.22560267668749840643223616905, 2.38116356225299620011633985249, 4.79085093103530693850477639954, 5.80045457911231193627474837008, 6.75395532990341575165448572479, 7.79298545455764963624452711488, 8.972019134073409502580254099142, 9.598150501746471856910168767253, 10.38252022530035955702643197808, 10.86340229289232847766758241404
