| L(s) = 1 | − 54·5-s + 88·7-s + 540·11-s − 418·13-s − 594·17-s − 836·19-s − 4.10e3·23-s − 209·25-s + 594·29-s − 4.25e3·31-s − 4.75e3·35-s − 298·37-s − 1.72e4·41-s + 1.21e4·43-s − 1.29e3·47-s − 9.06e3·49-s − 1.94e4·53-s − 2.91e4·55-s − 7.66e3·59-s − 3.47e4·61-s + 2.25e4·65-s − 2.18e4·67-s − 4.68e4·71-s + 6.75e4·73-s + 4.75e4·77-s + 7.69e4·79-s + 6.77e4·83-s + ⋯ |
| L(s) = 1 | − 0.965·5-s + 0.678·7-s + 1.34·11-s − 0.685·13-s − 0.498·17-s − 0.531·19-s − 1.61·23-s − 0.0668·25-s + 0.131·29-s − 0.795·31-s − 0.655·35-s − 0.0357·37-s − 1.60·41-s + 0.997·43-s − 0.0855·47-s − 0.539·49-s − 0.953·53-s − 1.29·55-s − 0.286·59-s − 1.19·61-s + 0.662·65-s − 0.593·67-s − 1.10·71-s + 1.48·73-s + 0.913·77-s + 1.38·79-s + 1.07·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 54 T + p^{5} T^{2} \) |
| 7 | \( 1 - 88 T + p^{5} T^{2} \) |
| 11 | \( 1 - 540 T + p^{5} T^{2} \) |
| 13 | \( 1 + 418 T + p^{5} T^{2} \) |
| 17 | \( 1 + 594 T + p^{5} T^{2} \) |
| 19 | \( 1 + 44 p T + p^{5} T^{2} \) |
| 23 | \( 1 + 4104 T + p^{5} T^{2} \) |
| 29 | \( 1 - 594 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4256 T + p^{5} T^{2} \) |
| 37 | \( 1 + 298 T + p^{5} T^{2} \) |
| 41 | \( 1 + 17226 T + p^{5} T^{2} \) |
| 43 | \( 1 - 12100 T + p^{5} T^{2} \) |
| 47 | \( 1 + 1296 T + p^{5} T^{2} \) |
| 53 | \( 1 + 19494 T + p^{5} T^{2} \) |
| 59 | \( 1 + 7668 T + p^{5} T^{2} \) |
| 61 | \( 1 + 34738 T + p^{5} T^{2} \) |
| 67 | \( 1 + 21812 T + p^{5} T^{2} \) |
| 71 | \( 1 + 46872 T + p^{5} T^{2} \) |
| 73 | \( 1 - 67562 T + p^{5} T^{2} \) |
| 79 | \( 1 - 76912 T + p^{5} T^{2} \) |
| 83 | \( 1 - 67716 T + p^{5} T^{2} \) |
| 89 | \( 1 + 29754 T + p^{5} T^{2} \) |
| 97 | \( 1 + 122398 T + p^{5} T^{2} \) |
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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
−11.78337473373533442945102350411, −10.86576828447727689229860365551, −9.528212767003365031346113091743, −8.408316367684768175408892340374, −7.50227109462292704894734035308, −6.30062336292508158626685161338, −4.65899818812156558486180855346, −3.73941500160716333790480641005, −1.81779474224752790939199077668, 0,
1.81779474224752790939199077668, 3.73941500160716333790480641005, 4.65899818812156558486180855346, 6.30062336292508158626685161338, 7.50227109462292704894734035308, 8.408316367684768175408892340374, 9.528212767003365031346113091743, 10.86576828447727689229860365551, 11.78337473373533442945102350411
