| L(s) = 1 | − 3-s − 7-s + 9-s − 11-s − 2·13-s − 2·17-s + 4·19-s + 21-s − 27-s − 6·29-s + 8·31-s + 33-s − 10·37-s + 2·39-s − 6·41-s + 12·43-s − 8·47-s + 49-s + 2·51-s − 10·53-s − 4·57-s + 12·59-s + 2·61-s − 63-s + 4·67-s + 6·73-s + 77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 0.218·21-s − 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.174·33-s − 1.64·37-s + 0.320·39-s − 0.937·41-s + 1.82·43-s − 1.16·47-s + 1/7·49-s + 0.280·51-s − 1.37·53-s − 0.529·57-s + 1.56·59-s + 0.256·61-s − 0.125·63-s + 0.488·67-s + 0.702·73-s + 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.788580598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.788580598\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−12.41672115512174, −12.02527139808551, −11.65490302459454, −11.16588147251847, −10.73142755105256, −10.20025505389225, −9.855157837512798, −9.431923085247322, −8.936753794756857, −8.377713595777168, −7.865795580377427, −7.340289913305689, −7.014422079365003, −6.427219627831821, −6.064655313567863, −5.437943285974911, −4.987652824542702, −4.696890440941150, −3.956550776691740, −3.371040264040093, −3.034851590534696, −2.086730690678469, −1.916168010113072, −0.8380472093687317, −0.4574559344203324,
0.4574559344203324, 0.8380472093687317, 1.916168010113072, 2.086730690678469, 3.034851590534696, 3.371040264040093, 3.956550776691740, 4.696890440941150, 4.987652824542702, 5.437943285974911, 6.064655313567863, 6.427219627831821, 7.014422079365003, 7.340289913305689, 7.865795580377427, 8.377713595777168, 8.936753794756857, 9.431923085247322, 9.855157837512798, 10.20025505389225, 10.73142755105256, 11.16588147251847, 11.65490302459454, 12.02527139808551, 12.41672115512174
