| L(s) = 1 | − 2·4-s + 5-s − 7-s − 5·13-s + 4·16-s + 3·17-s − 2·19-s − 2·20-s + 6·23-s + 25-s + 2·28-s + 3·29-s − 4·31-s − 35-s + 2·37-s − 12·41-s + 10·43-s − 9·47-s + 49-s + 10·52-s − 12·53-s − 8·61-s − 8·64-s − 5·65-s − 4·67-s − 6·68-s − 2·73-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 0.377·7-s − 1.38·13-s + 16-s + 0.727·17-s − 0.458·19-s − 0.447·20-s + 1.25·23-s + 1/5·25-s + 0.377·28-s + 0.557·29-s − 0.718·31-s − 0.169·35-s + 0.328·37-s − 1.87·41-s + 1.52·43-s − 1.31·47-s + 1/7·49-s + 1.38·52-s − 1.64·53-s − 1.02·61-s − 64-s − 0.620·65-s − 0.488·67-s − 0.727·68-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.101250267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.101250267\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.86680322150568, −14.26598467173504, −13.88850690004816, −13.13909702423378, −12.89826864598804, −12.32706584061824, −11.93084895401735, −11.05210553560681, −10.45857745627943, −10.00762404916466, −9.480336949375007, −9.140805422170037, −8.537897704517334, −7.808993752438724, −7.398205515982461, −6.637768999138422, −6.071870717061047, −5.339506968460512, −4.849722118730526, −4.498372675119965, −3.414354587964654, −3.120390900064215, −2.197307776479232, −1.335456603575525, −0.4081811529310983,
0.4081811529310983, 1.335456603575525, 2.197307776479232, 3.120390900064215, 3.414354587964654, 4.498372675119965, 4.849722118730526, 5.339506968460512, 6.071870717061047, 6.637768999138422, 7.398205515982461, 7.808993752438724, 8.537897704517334, 9.140805422170037, 9.480336949375007, 10.00762404916466, 10.45857745627943, 11.05210553560681, 11.93084895401735, 12.32706584061824, 12.89826864598804, 13.13909702423378, 13.88850690004816, 14.26598467173504, 14.86680322150568
