| L(s) = 1 | + 2-s + 4-s + 4·5-s + 7-s + 8-s + 4·10-s − 2·11-s + 2·13-s + 14-s + 16-s − 2·17-s − 2·19-s + 4·20-s − 2·22-s − 23-s + 11·25-s + 2·26-s + 28-s + 6·29-s + 32-s − 2·34-s + 4·35-s + 4·37-s − 2·38-s + 4·40-s + 10·41-s − 10·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.377·7-s + 0.353·8-s + 1.26·10-s − 0.603·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.458·19-s + 0.894·20-s − 0.426·22-s − 0.208·23-s + 11/5·25-s + 0.392·26-s + 0.188·28-s + 1.11·29-s + 0.176·32-s − 0.342·34-s + 0.676·35-s + 0.657·37-s − 0.324·38-s + 0.632·40-s + 1.56·41-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.310053153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.310053153\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.815068230456612616793582368352, −8.033720326381609393245153089068, −6.97024405714531274705574717557, −6.23546190177076507883290691713, −5.74970135644932715802309761153, −4.98074577986713805314334031227, −4.22286135037535152332353228728, −2.87069624217789131536586766161, −2.25059826043870520350019766587, −1.29083604794858859930694268758,
1.29083604794858859930694268758, 2.25059826043870520350019766587, 2.87069624217789131536586766161, 4.22286135037535152332353228728, 4.98074577986713805314334031227, 5.74970135644932715802309761153, 6.23546190177076507883290691713, 6.97024405714531274705574717557, 8.033720326381609393245153089068, 8.815068230456612616793582368352
