| L(s) = 1 | − 2-s + 3·3-s + 4-s + 3·5-s − 3·6-s − 8-s + 6·9-s − 3·10-s − 11-s + 3·12-s − 3·13-s + 9·15-s + 16-s + 4·17-s − 6·18-s + 8·19-s + 3·20-s + 22-s − 3·24-s + 4·25-s + 3·26-s + 9·27-s − 29-s − 9·30-s − 3·31-s − 32-s − 3·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s + 1.34·5-s − 1.22·6-s − 0.353·8-s + 2·9-s − 0.948·10-s − 0.301·11-s + 0.866·12-s − 0.832·13-s + 2.32·15-s + 1/4·16-s + 0.970·17-s − 1.41·18-s + 1.83·19-s + 0.670·20-s + 0.213·22-s − 0.612·24-s + 4/5·25-s + 0.588·26-s + 1.73·27-s − 0.185·29-s − 1.64·30-s − 0.538·31-s − 0.176·32-s − 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.361781767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.361781767\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−9.078793478982630504759051727737, −8.014059624755673577910993075273, −7.56184871379412699606892896514, −6.88813832101286898905501839843, −5.71806888345215349473669167955, −5.01700778132592015392719300838, −3.54554784662016365387565834933, −2.88346851767372352448305483512, −2.10420868538686379511979015792, −1.31406868181197915857989999912,
1.31406868181197915857989999912, 2.10420868538686379511979015792, 2.88346851767372352448305483512, 3.54554784662016365387565834933, 5.01700778132592015392719300838, 5.71806888345215349473669167955, 6.88813832101286898905501839843, 7.56184871379412699606892896514, 8.014059624755673577910993075273, 9.078793478982630504759051727737
