| L(s) = 1 | + 3-s + 5·7-s + 9-s + 11-s + 13-s − 3·17-s − 6·19-s + 5·21-s + 3·23-s + 27-s − 4·29-s + 33-s − 5·37-s + 39-s + 11·41-s − 6·43-s + 18·49-s − 3·51-s − 9·53-s − 6·57-s + 12·59-s − 5·61-s + 5·63-s − 8·67-s + 3·69-s − 13·71-s + 10·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.88·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s − 0.727·17-s − 1.37·19-s + 1.09·21-s + 0.625·23-s + 0.192·27-s − 0.742·29-s + 0.174·33-s − 0.821·37-s + 0.160·39-s + 1.71·41-s − 0.914·43-s + 18/7·49-s − 0.420·51-s − 1.23·53-s − 0.794·57-s + 1.56·59-s − 0.640·61-s + 0.629·63-s − 0.977·67-s + 0.361·69-s − 1.54·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.116391512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.116391512\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.29296037475548, −13.92372948862723, −13.28865413579407, −12.84805961046407, −12.28952555932352, −11.55067857507138, −11.28081808993444, −10.69140644788222, −10.44434858805847, −9.491337714461772, −8.894889161874658, −8.713440012216363, −8.042433727302544, −7.704510955909513, −7.044585607855388, −6.478765303633298, −5.784697791535434, −5.092287674507162, −4.556740947446235, −4.178534581656145, −3.495827634334591, −2.600777385171781, −1.927507759227821, −1.624849732964532, −0.6561816101048871,
0.6561816101048871, 1.624849732964532, 1.927507759227821, 2.600777385171781, 3.495827634334591, 4.178534581656145, 4.556740947446235, 5.092287674507162, 5.784697791535434, 6.478765303633298, 7.044585607855388, 7.704510955909513, 8.042433727302544, 8.713440012216363, 8.894889161874658, 9.491337714461772, 10.44434858805847, 10.69140644788222, 11.28081808993444, 11.55067857507138, 12.28952555932352, 12.84805961046407, 13.28865413579407, 13.92372948862723, 14.29296037475548
