| L(s) = 1 | + 34·5-s − 240·7-s + 124·11-s + 46·13-s − 1.95e3·17-s − 1.92e3·19-s − 2.84e3·23-s − 1.96e3·25-s + 8.92e3·29-s − 4.64e3·31-s − 8.16e3·35-s − 4.36e3·37-s + 2.88e3·41-s + 1.13e4·43-s − 7.00e3·47-s + 4.07e4·49-s + 2.25e4·53-s + 4.21e3·55-s + 28·59-s − 6.38e3·61-s + 1.56e3·65-s − 3.90e4·67-s + 5.48e4·71-s + 2.10e4·73-s − 2.97e4·77-s + 2.66e4·79-s − 5.61e4·83-s + ⋯ |
| L(s) = 1 | + 0.608·5-s − 1.85·7-s + 0.308·11-s + 0.0754·13-s − 1.63·17-s − 1.22·19-s − 1.11·23-s − 0.630·25-s + 1.97·29-s − 0.868·31-s − 1.12·35-s − 0.523·37-s + 0.268·41-s + 0.934·43-s − 0.462·47-s + 2.42·49-s + 1.10·53-s + 0.187·55-s + 0.00104·59-s − 0.219·61-s + 0.0459·65-s − 1.06·67-s + 1.29·71-s + 0.461·73-s − 0.572·77-s + 0.480·79-s − 0.895·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 34 T + p^{5} T^{2} \) |
| 7 | \( 1 + 240 T + p^{5} T^{2} \) |
| 11 | \( 1 - 124 T + p^{5} T^{2} \) |
| 13 | \( 1 - 46 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1954 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1924 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2840 T + p^{5} T^{2} \) |
| 29 | \( 1 - 8922 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4648 T + p^{5} T^{2} \) |
| 37 | \( 1 + 4362 T + p^{5} T^{2} \) |
| 41 | \( 1 - 2886 T + p^{5} T^{2} \) |
| 43 | \( 1 - 11332 T + p^{5} T^{2} \) |
| 47 | \( 1 + 7008 T + p^{5} T^{2} \) |
| 53 | \( 1 - 22594 T + p^{5} T^{2} \) |
| 59 | \( 1 - 28 T + p^{5} T^{2} \) |
| 61 | \( 1 + 6386 T + p^{5} T^{2} \) |
| 67 | \( 1 + 39076 T + p^{5} T^{2} \) |
| 71 | \( 1 - 54872 T + p^{5} T^{2} \) |
| 73 | \( 1 - 21034 T + p^{5} T^{2} \) |
| 79 | \( 1 - 26632 T + p^{5} T^{2} \) |
| 83 | \( 1 + 56188 T + p^{5} T^{2} \) |
| 89 | \( 1 + 64410 T + p^{5} T^{2} \) |
| 97 | \( 1 + 116158 T + p^{5} T^{2} \) |
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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
−13.14208723678942903620643397804, −12.24679800166897611911176176575, −10.63275974819259815133781873469, −9.690266573201177113947341043907, −8.712802837188817284206805893915, −6.74686597021038250976163478928, −6.07976192635138161020279602619, −4.04834312181204936664586322353, −2.39895315587275430503900070191, 0,
2.39895315587275430503900070191, 4.04834312181204936664586322353, 6.07976192635138161020279602619, 6.74686597021038250976163478928, 8.712802837188817284206805893915, 9.690266573201177113947341043907, 10.63275974819259815133781873469, 12.24679800166897611911176176575, 13.14208723678942903620643397804
