| L(s) = 1 | − 9·9-s + 60·11-s + 298·19-s + 468·29-s + 434·31-s − 312·41-s + 157·49-s − 540·59-s + 550·61-s − 1.32e3·71-s + 1.98e3·79-s + 81·81-s + 2.97e3·89-s − 540·99-s − 1.58e3·101-s + 110·109-s + 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.55e3·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.64·11-s + 3.59·19-s + 2.99·29-s + 2.51·31-s − 1.18·41-s + 0.457·49-s − 1.19·59-s + 1.15·61-s − 2.20·71-s + 2.82·79-s + 1/9·81-s + 3.54·89-s − 0.548·99-s − 1.56·101-s + 0.0966·109-s + 0.0285·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.61·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.022405805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.022405805\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 157 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 30 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3553 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3742 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 149 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1834 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 234 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 p T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 79990 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 156 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 28475 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 206746 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6950 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 270 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 275 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 43283 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 660 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 360718 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 992 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 427858 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1488 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1723585 T^{2} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.470790463770684221813329414928, −9.249577113901264539004440589836, −8.870337955392593314740584989692, −8.196902246826663031048149774595, −8.100817887404686094556285228167, −7.57317961276147072413263548143, −6.87522044329591554760566656315, −6.83410563506039453038955461420, −6.17008672141636631960439611969, −6.02049436124790095979914530722, −5.12781576271603856965176310994, −4.96989512164498237753778498928, −4.51988345346468985305412357483, −3.84498265680341999375986738709, −3.20553261783089610249821696390, −3.09055219644516106093369298991, −2.44861461137075857299241436782, −1.45386534155897433924503298068, −0.926552399898702445190354227642, −0.817735468916531309451112835020,
0.817735468916531309451112835020, 0.926552399898702445190354227642, 1.45386534155897433924503298068, 2.44861461137075857299241436782, 3.09055219644516106093369298991, 3.20553261783089610249821696390, 3.84498265680341999375986738709, 4.51988345346468985305412357483, 4.96989512164498237753778498928, 5.12781576271603856965176310994, 6.02049436124790095979914530722, 6.17008672141636631960439611969, 6.83410563506039453038955461420, 6.87522044329591554760566656315, 7.57317961276147072413263548143, 8.100817887404686094556285228167, 8.196902246826663031048149774595, 8.870337955392593314740584989692, 9.249577113901264539004440589836, 9.470790463770684221813329414928
