| L(s) = 1 | + 39.9·5-s + 85.8·7-s − 240.·11-s + 972.·25-s + 818·29-s + 1.37e3·31-s + 3.43e3·35-s + 4.96e3·49-s + 2.37e3·53-s − 9.62e3·55-s + 6.86e3·59-s − 1.86e3·73-s − 2.06e4·77-s + 9.11e3·79-s − 9.28e3·83-s − 1.50e4·97-s − 1.91e4·101-s + 2.11e4·103-s + 6.84e3·107-s + ⋯ |
| L(s) = 1 | + 1.59·5-s + 1.75·7-s − 1.98·11-s + 1.55·25-s + 0.972·29-s + 1.42·31-s + 2.80·35-s + 2.06·49-s + 0.846·53-s − 3.18·55-s + 1.97·59-s − 0.349·73-s − 3.48·77-s + 1.46·79-s − 1.34·83-s − 1.60·97-s − 1.87·101-s + 1.99·103-s + 0.597·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.628646264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.628646264\) |
| \(L(3)\) |
|
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 | \( 1 - 39.9T + 625T^{2} \) |
| 7 | \( 1 - 85.8T + 2.40e3T^{2} \) |
| 11 | \( 1 + 240.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 2.85e4T^{2} \) |
| 17 | \( 1 - 8.35e4T^{2} \) |
| 19 | \( 1 - 1.30e5T^{2} \) |
| 23 | \( 1 - 2.79e5T^{2} \) |
| 29 | \( 1 - 818T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.37e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.41e6T^{2} \) |
| 47 | \( 1 - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.37e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 6.86e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 1.38e7T^{2} \) |
| 67 | \( 1 - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.86e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 9.11e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 9.28e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.50e4T + 8.85e7T^{2} \) |
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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
−9.829764864808069207274190446632, −8.536442948149204994251704669904, −8.119699425625088767432076833710, −7.06404981699042072104428668652, −5.83725436190528653921161347098, −5.22483371581731377778526614329, −4.60563858874137377981076889933, −2.67797507150908905423887936786, −2.09196090254101209320707458005, −0.977444546020724471232171683401,
0.977444546020724471232171683401, 2.09196090254101209320707458005, 2.67797507150908905423887936786, 4.60563858874137377981076889933, 5.22483371581731377778526614329, 5.83725436190528653921161347098, 7.06404981699042072104428668652, 8.119699425625088767432076833710, 8.536442948149204994251704669904, 9.829764864808069207274190446632
