L(s) = 1 | + 3-s + 7-s + 9-s − 4·11-s + 6·13-s + 6·17-s − 4·19-s + 21-s + 8·23-s + 27-s − 10·29-s + 4·31-s − 4·33-s − 6·37-s + 6·39-s + 6·41-s + 4·43-s + 12·47-s + 49-s + 6·51-s + 6·53-s − 4·57-s − 4·59-s + 2·61-s + 63-s + 4·67-s + 8·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 1.45·17-s − 0.917·19-s + 0.218·21-s + 1.66·23-s + 0.192·27-s − 1.85·29-s + 0.718·31-s − 0.696·33-s − 0.986·37-s + 0.960·39-s + 0.937·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s − 0.529·57-s − 0.520·59-s + 0.256·61-s + 0.125·63-s + 0.488·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.619175596\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.619175596\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−15.18355568987610, −14.36734585983601, −14.03491367827058, −13.33380470452342, −12.98024399551107, −12.60252432365287, −11.80812109007902, −11.11850708225857, −10.69354702297703, −10.39019659021491, −9.557055720455754, −8.908811082797586, −8.597747326490212, −7.941468427953258, −7.476845537062825, −6.969287019797171, −5.982309335136236, −5.624557307939741, −5.008354063842486, −4.141213359662499, −3.627442902407994, −2.965234332380283, −2.300561509433939, −1.434490586912761, −0.7363448787142799,
0.7363448787142799, 1.434490586912761, 2.300561509433939, 2.965234332380283, 3.627442902407994, 4.141213359662499, 5.008354063842486, 5.624557307939741, 5.982309335136236, 6.969287019797171, 7.476845537062825, 7.941468427953258, 8.597747326490212, 8.908811082797586, 9.557055720455754, 10.39019659021491, 10.69354702297703, 11.11850708225857, 11.80812109007902, 12.60252432365287, 12.98024399551107, 13.33380470452342, 14.03491367827058, 14.36734585983601, 15.18355568987610