| L(s) = 1 | − 9·3-s + 94·5-s + 144·7-s + 81·9-s − 380·11-s + 814·13-s − 846·15-s − 862·17-s − 1.15e3·19-s − 1.29e3·21-s − 488·23-s + 5.71e3·25-s − 729·27-s − 5.46e3·29-s + 9.56e3·31-s + 3.42e3·33-s + 1.35e4·35-s − 1.05e4·37-s − 7.32e3·39-s − 5.19e3·41-s − 1.70e4·43-s + 7.61e3·45-s + 3.16e3·47-s + 3.92e3·49-s + 7.75e3·51-s − 2.47e4·53-s − 3.57e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.68·5-s + 1.11·7-s + 1/3·9-s − 0.946·11-s + 1.33·13-s − 0.970·15-s − 0.723·17-s − 0.734·19-s − 0.641·21-s − 0.192·23-s + 1.82·25-s − 0.192·27-s − 1.20·29-s + 1.78·31-s + 0.546·33-s + 1.86·35-s − 1.26·37-s − 0.771·39-s − 0.482·41-s − 1.40·43-s + 0.560·45-s + 0.209·47-s + 0.233·49-s + 0.417·51-s − 1.21·53-s − 1.59·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.622509299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.622509299\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 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 | \( 1 + 16318 T + p^{5} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.02369362686041213188635816632, −15.46322260315241086307427460748, −13.90561383579375957615390350628, −13.06940221160175180688723339711, −11.19002072297723253460719666808, −10.20366154866776066726332962923, −8.507946641379563185979674713595, −6.32580536826711312154384718220, −5.08043612857642256105006011169, −1.80248271161811922299803438914,
1.80248271161811922299803438914, 5.08043612857642256105006011169, 6.32580536826711312154384718220, 8.507946641379563185979674713595, 10.20366154866776066726332962923, 11.19002072297723253460719666808, 13.06940221160175180688723339711, 13.90561383579375957615390350628, 15.46322260315241086307427460748, 17.02369362686041213188635816632
