L(s) = 1 | − 4·3-s + 4-s + 6·9-s − 4·12-s − 8·13-s + 16-s + 4·19-s − 10·25-s + 4·27-s + 6·36-s + 32·39-s + 12·41-s − 24·47-s − 4·48-s + 49-s − 8·52-s − 16·57-s + 8·61-s + 64-s + 4·73-s + 40·75-s + 4·76-s − 37·81-s − 12·83-s − 20·97-s − 10·100-s − 8·103-s + ⋯ |
L(s) = 1 | − 2.30·3-s + 1/2·4-s + 2·9-s − 1.15·12-s − 2.21·13-s + 1/4·16-s + 0.917·19-s − 2·25-s + 0.769·27-s + 36-s + 5.12·39-s + 1.87·41-s − 3.50·47-s − 0.577·48-s + 1/7·49-s − 1.10·52-s − 2.11·57-s + 1.02·61-s + 1/8·64-s + 0.468·73-s + 4.61·75-s + 0.458·76-s − 4.11·81-s − 1.31·83-s − 2.03·97-s − 100-s − 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2513182106\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2513182106\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 61 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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
−8.173449077566698438751222908068, −7.57571100088867902110310233811, −7.33566132154895538455272247406, −6.86227272075079610412028702533, −6.34728331211217856949764881281, −6.01523816258371890896044883784, −5.57928681742950427486583645839, −5.17900673049925304081269440708, −4.85492146999244580615110535351, −4.38080265430113012265676461865, −3.56574648496591683143026739238, −2.79745735406815554125194503256, −2.27907197519064804199458946410, −1.34430444523648923531944095258, −0.28992103454452119940322760112,
0.28992103454452119940322760112, 1.34430444523648923531944095258, 2.27907197519064804199458946410, 2.79745735406815554125194503256, 3.56574648496591683143026739238, 4.38080265430113012265676461865, 4.85492146999244580615110535351, 5.17900673049925304081269440708, 5.57928681742950427486583645839, 6.01523816258371890896044883784, 6.34728331211217856949764881281, 6.86227272075079610412028702533, 7.33566132154895538455272247406, 7.57571100088867902110310233811, 8.173449077566698438751222908068