| L(s) = 1 | − 88.1·5-s + 29·7-s − 88.1·11-s + 329·13-s + 2.20e3·17-s + 1.79e3·19-s − 3.61e3·23-s + 4.65e3·25-s − 1.41e3·29-s + 5.22e3·31-s − 2.55e3·35-s + 8.78e3·37-s + 1.55e4·41-s + 1.99e4·43-s + 1.08e4·47-s − 1.59e4·49-s − 2.94e4·53-s + 7.77e3·55-s − 5.73e3·59-s − 1.06e3·61-s − 2.90e4·65-s − 6.20e4·67-s + 4.63e4·71-s − 4.80e4·73-s − 2.55e3·77-s + 4.99e4·79-s + 5.76e4·83-s + ⋯ |
| L(s) = 1 | − 1.57·5-s + 0.223·7-s − 0.219·11-s + 0.539·13-s + 1.85·17-s + 1.14·19-s − 1.42·23-s + 1.48·25-s − 0.311·29-s + 0.977·31-s − 0.352·35-s + 1.05·37-s + 1.44·41-s + 1.64·43-s + 0.716·47-s − 0.949·49-s − 1.44·53-s + 0.346·55-s − 0.214·59-s − 0.0367·61-s − 0.851·65-s − 1.68·67-s + 1.09·71-s − 1.05·73-s − 0.0491·77-s + 0.900·79-s + 0.918·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.399706823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.399706823\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 88.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 29T + 1.68e4T^{2} \) |
| 11 | \( 1 + 88.1T + 1.61e5T^{2} \) |
| 13 | \( 1 - 329T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.20e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.79e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.61e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.22e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.78e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.55e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.99e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.08e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.94e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.73e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.06e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.20e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.63e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.80e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.99e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.76e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.79e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.29e4T + 8.58e9T^{2} \) |
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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
−12.42439022651627505476971439061, −11.81780717363408966068934463178, −10.84350465638465981989280208824, −9.539194038455000706818034120299, −7.929420413152470749238584238208, −7.69214051862478737708636673214, −5.85762870602224772323332024007, −4.32387649703875644622264018298, −3.21335658878699340620447031106, −0.863284644929729163948244231784,
0.863284644929729163948244231784, 3.21335658878699340620447031106, 4.32387649703875644622264018298, 5.85762870602224772323332024007, 7.69214051862478737708636673214, 7.929420413152470749238584238208, 9.539194038455000706818034120299, 10.84350465638465981989280208824, 11.81780717363408966068934463178, 12.42439022651627505476971439061
