| L(s) = 1 | + 32·2-s + 1.02e3·4-s − 3.12e3·5-s − 2.68e4·7-s + 3.27e4·8-s + 5.90e4·9-s − 1.00e5·10-s + 3.21e5·11-s + 6.82e5·13-s − 8.60e5·14-s + 1.04e6·16-s + 1.88e6·18-s − 1.28e6·19-s − 3.20e6·20-s + 1.02e7·22-s + 1.77e6·23-s + 9.76e6·25-s + 2.18e7·26-s − 2.75e7·28-s + 3.35e7·32-s + 8.40e7·35-s + 6.04e7·36-s − 1.12e8·37-s − 4.12e7·38-s − 1.02e8·40-s − 6.11e5·41-s + 3.28e8·44-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 5-s − 1.59·7-s + 8-s + 9-s − 10-s + 1.99·11-s + 1.83·13-s − 1.59·14-s + 16-s + 18-s − 0.520·19-s − 20-s + 1.99·22-s + 0.275·23-s + 25-s + 1.83·26-s − 1.59·28-s + 32-s + 1.59·35-s + 36-s − 1.62·37-s − 0.520·38-s − 40-s − 0.00527·41-s + 1.99·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(3.484396828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.484396828\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{5} T \) |
| 5 | \( 1 + p^{5} T \) |
| good | 3 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 7 | \( 1 + 26886 T + p^{10} T^{2} \) |
| 11 | \( 1 - 321102 T + p^{10} T^{2} \) |
| 13 | \( 1 - 682086 T + p^{10} T^{2} \) |
| 17 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 19 | \( 1 + 1288802 T + p^{10} T^{2} \) |
| 23 | \( 1 - 1772186 T + p^{10} T^{2} \) |
| 29 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 31 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 37 | \( 1 + 112652586 T + p^{10} T^{2} \) |
| 41 | \( 1 + 611598 T + p^{10} T^{2} \) |
| 43 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 47 | \( 1 - 222705514 T + p^{10} T^{2} \) |
| 53 | \( 1 - 552886486 T + p^{10} T^{2} \) |
| 59 | \( 1 + 646632402 T + p^{10} T^{2} \) |
| 61 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 67 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 71 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 73 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 79 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 83 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 89 | \( 1 - 5417714898 T + p^{10} T^{2} \) |
| 97 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
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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
−13.76553475858875280288932855035, −12.73946846592560551657269579053, −11.86897117581987799054642231443, −10.58660702724215720709017805445, −8.945206105222637012804815076152, −6.97087349537319414664087226827, −6.31427518913810964724375156592, −4.00679183679950424192685361831, −3.55651034573834383959461642706, −1.17161597705916422204989183659,
1.17161597705916422204989183659, 3.55651034573834383959461642706, 4.00679183679950424192685361831, 6.31427518913810964724375156592, 6.97087349537319414664087226827, 8.945206105222637012804815076152, 10.58660702724215720709017805445, 11.86897117581987799054642231443, 12.73946846592560551657269579053, 13.76553475858875280288932855035
