| L(s) = 1 | − 26.3·5-s + 63.2·7-s + 98.2·11-s − 738.·13-s + 250.·17-s + 1.10e3·19-s + 4.40e3·23-s − 2.43e3·25-s − 7.88e3·29-s − 4.61e3·31-s − 1.66e3·35-s + 1.18e4·37-s + 1.00e4·41-s + 7.03e3·43-s + 1.49e4·47-s − 1.28e4·49-s + 2.24e4·53-s − 2.58e3·55-s + 1.08e4·59-s − 1.18e3·61-s + 1.94e4·65-s + 5.91e4·67-s + 1.43e4·71-s − 5.30e4·73-s + 6.21e3·77-s − 3.73e4·79-s + 1.20e5·83-s + ⋯ |
| L(s) = 1 | − 0.470·5-s + 0.487·7-s + 0.244·11-s − 1.21·13-s + 0.209·17-s + 0.700·19-s + 1.73·23-s − 0.778·25-s − 1.74·29-s − 0.861·31-s − 0.229·35-s + 1.42·37-s + 0.936·41-s + 0.580·43-s + 0.985·47-s − 0.761·49-s + 1.09·53-s − 0.115·55-s + 0.404·59-s − 0.0409·61-s + 0.570·65-s + 1.61·67-s + 0.337·71-s − 1.16·73-s + 0.119·77-s − 0.674·79-s + 1.92·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.830673089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.830673089\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 26.3T + 3.12e3T^{2} \) |
| 7 | \( 1 - 63.2T + 1.68e4T^{2} \) |
| 11 | \( 1 - 98.2T + 1.61e5T^{2} \) |
| 13 | \( 1 + 738.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 250.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.10e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.40e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.88e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.61e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.18e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.00e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.03e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.49e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.24e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.08e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.18e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.91e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.43e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.30e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.73e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.20e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.78e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.06e5T + 8.58e9T^{2} \) |
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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
−11.00677833580793825101460831867, −9.716179057559618794857136811480, −9.021291736564642204547510242027, −7.65193200083286467665377216040, −7.28177975772318231907714126292, −5.72599533346898860131356144993, −4.77426896484820120802894635640, −3.61539680447510848330928284998, −2.25872141062757164552752375287, −0.75104466199822278326154948823,
0.75104466199822278326154948823, 2.25872141062757164552752375287, 3.61539680447510848330928284998, 4.77426896484820120802894635640, 5.72599533346898860131356144993, 7.28177975772318231907714126292, 7.65193200083286467665377216040, 9.021291736564642204547510242027, 9.716179057559618794857136811480, 11.00677833580793825101460831867
