| L(s) = 1 | + 94·5-s + 144·7-s − 380·11-s − 814·13-s + 862·17-s + 1.15e3·19-s + 488·23-s + 5.71e3·25-s − 5.46e3·29-s + 9.56e3·31-s + 1.35e4·35-s + 1.05e4·37-s + 5.19e3·41-s + 1.70e4·43-s − 3.16e3·47-s + 3.92e3·49-s − 2.47e4·53-s − 3.57e4·55-s + 1.73e4·59-s − 4.36e3·61-s − 7.65e4·65-s + 5.28e3·67-s − 8.36e3·71-s + 3.94e4·73-s − 5.47e4·77-s + 4.23e4·79-s − 6.18e4·83-s + ⋯ |
| L(s) = 1 | + 1.68·5-s + 1.11·7-s − 0.946·11-s − 1.33·13-s + 0.723·17-s + 0.734·19-s + 0.192·23-s + 1.82·25-s − 1.20·29-s + 1.78·31-s + 1.86·35-s + 1.26·37-s + 0.482·41-s + 1.40·43-s − 0.209·47-s + 0.233·49-s − 1.21·53-s − 1.59·55-s + 0.650·59-s − 0.150·61-s − 2.24·65-s + 0.143·67-s − 0.196·71-s + 0.866·73-s − 1.05·77-s + 0.763·79-s − 0.985·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.524674734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.524674734\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 94 T + p^{5} T^{2} \) |
| 7 | \( 1 - 144 T + p^{5} T^{2} \) |
| 11 | \( 1 + 380 T + p^{5} T^{2} \) |
| 13 | \( 1 + 814 T + p^{5} T^{2} \) |
| 17 | \( 1 - 862 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1156 T + p^{5} T^{2} \) |
| 23 | \( 1 - 488 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5466 T + p^{5} T^{2} \) |
| 31 | \( 1 - 9560 T + p^{5} T^{2} \) |
| 37 | \( 1 - 10506 T + p^{5} T^{2} \) |
| 41 | \( 1 - 5190 T + p^{5} T^{2} \) |
| 43 | \( 1 - 17084 T + p^{5} T^{2} \) |
| 47 | \( 1 + 3168 T + p^{5} T^{2} \) |
| 53 | \( 1 + 24770 T + p^{5} T^{2} \) |
| 59 | \( 1 - 17380 T + p^{5} T^{2} \) |
| 61 | \( 1 + 4366 T + p^{5} T^{2} \) |
| 67 | \( 1 - 5284 T + p^{5} T^{2} \) |
| 71 | \( 1 + 8360 T + p^{5} T^{2} \) |
| 73 | \( 1 - 39466 T + p^{5} T^{2} \) |
| 79 | \( 1 - 42376 T + p^{5} T^{2} \) |
| 83 | \( 1 + 61828 T + p^{5} T^{2} \) |
| 89 | \( 1 - 63078 T + p^{5} T^{2} \) |
| 97 | \( 1 + 16318 T + p^{5} T^{2} \) |
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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
−9.846468371493817794712589438769, −9.366016020789870552713700693351, −8.048228944685122593852245416420, −7.41127961572257241598574164970, −6.08691176610031953317084897770, −5.30132052513081198376045526617, −4.70988476727441681127154916865, −2.79650012778167547279819888719, −2.07862216642059462641378252727, −0.955382484018132189083166124308,
0.955382484018132189083166124308, 2.07862216642059462641378252727, 2.79650012778167547279819888719, 4.70988476727441681127154916865, 5.30132052513081198376045526617, 6.08691176610031953317084897770, 7.41127961572257241598574164970, 8.048228944685122593852245416420, 9.366016020789870552713700693351, 9.846468371493817794712589438769
