| L(s) = 1 | − 27·3-s − 26·5-s + 1.05e3·7-s + 729·9-s + 6.41e3·11-s + 5.20e3·13-s + 702·15-s − 6.23e3·17-s + 4.14e4·19-s − 2.85e4·21-s − 2.94e4·23-s − 7.74e4·25-s − 1.96e4·27-s − 2.10e5·29-s + 1.85e5·31-s − 1.73e5·33-s − 2.74e4·35-s + 5.07e5·37-s − 1.40e5·39-s + 3.60e5·41-s + 6.20e5·43-s − 1.89e4·45-s − 8.47e5·47-s + 2.91e5·49-s + 1.68e5·51-s + 1.42e6·53-s − 1.66e5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.0930·5-s + 1.16·7-s + 1/3·9-s + 1.45·11-s + 0.657·13-s + 0.0537·15-s − 0.307·17-s + 1.38·19-s − 0.671·21-s − 0.504·23-s − 0.991·25-s − 0.192·27-s − 1.60·29-s + 1.11·31-s − 0.838·33-s − 0.108·35-s + 1.64·37-s − 0.379·39-s + 0.815·41-s + 1.18·43-s − 0.0310·45-s − 1.19·47-s + 0.354·49-s + 0.177·51-s + 1.31·53-s − 0.135·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.618334078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.618334078\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{3} T \) |
| good | 5 | \( 1 + 26 T + p^{7} T^{2} \) |
| 7 | \( 1 - 1056 T + p^{7} T^{2} \) |
| 11 | \( 1 - 6412 T + p^{7} T^{2} \) |
| 13 | \( 1 - 5206 T + p^{7} T^{2} \) |
| 17 | \( 1 + 6238 T + p^{7} T^{2} \) |
| 19 | \( 1 - 41492 T + p^{7} T^{2} \) |
| 23 | \( 1 + 29432 T + p^{7} T^{2} \) |
| 29 | \( 1 + 210498 T + p^{7} T^{2} \) |
| 31 | \( 1 - 185240 T + p^{7} T^{2} \) |
| 37 | \( 1 - 507630 T + p^{7} T^{2} \) |
| 41 | \( 1 - 360042 T + p^{7} T^{2} \) |
| 43 | \( 1 - 620044 T + p^{7} T^{2} \) |
| 47 | \( 1 + 847680 T + p^{7} T^{2} \) |
| 53 | \( 1 - 1423750 T + p^{7} T^{2} \) |
| 59 | \( 1 + 2548724 T + p^{7} T^{2} \) |
| 61 | \( 1 + 706058 T + p^{7} T^{2} \) |
| 67 | \( 1 + 2418796 T + p^{7} T^{2} \) |
| 71 | \( 1 - 265976 T + p^{7} T^{2} \) |
| 73 | \( 1 + 5791238 T + p^{7} T^{2} \) |
| 79 | \( 1 - 2955688 T + p^{7} T^{2} \) |
| 83 | \( 1 - 3462932 T + p^{7} T^{2} \) |
| 89 | \( 1 + 2211126 T + p^{7} T^{2} \) |
| 97 | \( 1 + 15594814 T + p^{7} T^{2} \) |
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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
−16.25663400719246772264176036475, −14.85297163848349863264819367341, −13.68074608558994041148286555720, −11.86414974782837222992271368736, −11.20271569300647999154752345814, −9.387201354387064193135600659112, −7.70793143058638113750395474599, −5.95595021113712118782826221811, −4.21357909420311406672017984735, −1.32447400673339297271802058052,
1.32447400673339297271802058052, 4.21357909420311406672017984735, 5.95595021113712118782826221811, 7.70793143058638113750395474599, 9.387201354387064193135600659112, 11.20271569300647999154752345814, 11.86414974782837222992271368736, 13.68074608558994041148286555720, 14.85297163848349863264819367341, 16.25663400719246772264176036475
