L(s) = 1 | − 3-s + 4·5-s + 3·7-s − 2·9-s + 5·11-s − 4·15-s − 6·17-s + 2·19-s − 3·21-s + 6·23-s + 11·25-s + 5·27-s + 6·29-s − 4·31-s − 5·33-s + 12·35-s − 37-s − 9·41-s + 4·43-s − 8·45-s + 7·47-s + 2·49-s + 6·51-s − 9·53-s + 20·55-s − 2·57-s − 4·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.78·5-s + 1.13·7-s − 2/3·9-s + 1.50·11-s − 1.03·15-s − 1.45·17-s + 0.458·19-s − 0.654·21-s + 1.25·23-s + 11/5·25-s + 0.962·27-s + 1.11·29-s − 0.718·31-s − 0.870·33-s + 2.02·35-s − 0.164·37-s − 1.40·41-s + 0.609·43-s − 1.19·45-s + 1.02·47-s + 2/7·49-s + 0.840·51-s − 1.23·53-s + 2.69·55-s − 0.264·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.495025734\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.495025734\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.857051213666405733950065270441, −8.662999064798443542402998841039, −7.13510933071835526496116929392, −6.49721024043084796304110532521, −5.88701276732509731326794401981, −5.08525742835756647812395813836, −4.51548276160158828731253741798, −2.99215513529564679901126733295, −1.93477151442205638399476972753, −1.17108677221333076846883554267,
1.17108677221333076846883554267, 1.93477151442205638399476972753, 2.99215513529564679901126733295, 4.51548276160158828731253741798, 5.08525742835756647812395813836, 5.88701276732509731326794401981, 6.49721024043084796304110532521, 7.13510933071835526496116929392, 8.662999064798443542402998841039, 8.857051213666405733950065270441