| L(s) = 1 | − 3-s + 2·5-s + 7-s + 9-s + 2·13-s − 2·15-s + 4·17-s + 4·19-s − 21-s + 6·23-s − 25-s − 27-s + 2·29-s + 2·31-s + 2·35-s + 2·37-s − 2·39-s + 6·43-s + 2·45-s + 12·47-s + 49-s − 4·51-s − 4·57-s + 12·59-s − 2·61-s + 63-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s + 0.970·17-s + 0.917·19-s − 0.218·21-s + 1.25·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.359·31-s + 0.338·35-s + 0.328·37-s − 0.320·39-s + 0.914·43-s + 0.298·45-s + 1.75·47-s + 1/7·49-s − 0.560·51-s − 0.529·57-s + 1.56·59-s − 0.256·61-s + 0.125·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.531206787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.531206787\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.73888934952374, −14.07652730730926, −13.77947920534959, −13.30849282984275, −12.56746513090065, −12.26882538014637, −11.51695058056515, −11.18754814355047, −10.48703874667039, −10.16208105608377, −9.424701805170885, −9.157380083056939, −8.348751705434826, −7.744762485176304, −7.200157424375782, −6.563007687845769, −5.962869098670420, −5.412613788319559, −5.149320698019464, −4.269307307548977, −3.613531806306899, −2.811761145210536, −2.141470450051741, −1.161474273159924, −0.8632771452444050,
0.8632771452444050, 1.161474273159924, 2.141470450051741, 2.811761145210536, 3.613531806306899, 4.269307307548977, 5.149320698019464, 5.412613788319559, 5.962869098670420, 6.563007687845769, 7.200157424375782, 7.744762485176304, 8.348751705434826, 9.157380083056939, 9.424701805170885, 10.16208105608377, 10.48703874667039, 11.18754814355047, 11.51695058056515, 12.26882538014637, 12.56746513090065, 13.30849282984275, 13.77947920534959, 14.07652730730926, 14.73888934952374
