| L(s) = 1 | + 192·13-s + 2.49e3·25-s + 5.17e3·37-s + 4.22e3·49-s + 1.36e4·61-s + 3.60e4·73-s + 4.03e4·97-s − 3.78e4·109-s + 3.09e4·121-s + ⋯ |
| L(s) = 1 | + 1.13·13-s + 3.99·25-s + 3.78·37-s + 1.75·49-s + 3.66·61-s + 6.77·73-s + 4.28·97-s − 3.18·109-s + 2.11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(11.10990672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.10990672\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 1248 T^{2} + p^{8} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 2110 T^{2} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 15458 T^{2} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 48 T + p^{4} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 162624 T^{2} + p^{8} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 150050 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 214082 T^{2} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 753312 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 1784834 T^{2} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 1294 T + p^{4} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 4745664 T^{2} + p^{8} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 2165090 T^{2} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 3525502 T^{2} + p^{8} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 6144480 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 4272866 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 3410 T + p^{4} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 17050274 T^{2} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 9005762 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9024 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 74851970 T^{2} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 51742174 T^{2} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 106028160 T^{2} + p^{8} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10080 T + p^{4} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.202431150020850179598005786573, −7.51493357631170063068452219876, −7.33810669535277704967045103559, −7.20014988059104439849118628959, −6.93943179042441085574713281106, −6.34257202001605565835173484535, −6.32578571093637117320621063779, −6.32462612315227252053908141126, −5.93698541688893253148274222630, −5.39617893512852430428358896371, −4.98978336173385086920000277892, −4.94222407077986346501069941146, −4.94151141551248101791502101468, −4.29635119377844893824061518262, −3.85266923056098866677496472841, −3.64966428357987282931273495834, −3.63193578165173078636820807813, −2.95333111962387765731102884200, −2.50884402765617066878796630227, −2.42124277654336834283344548597, −2.20850656757112326361309090079, −1.25483055067858214271284339908, −0.906571772027680314112731932255, −0.880568028480232527772146548407, −0.58953290669076848577202707066,
0.58953290669076848577202707066, 0.880568028480232527772146548407, 0.906571772027680314112731932255, 1.25483055067858214271284339908, 2.20850656757112326361309090079, 2.42124277654336834283344548597, 2.50884402765617066878796630227, 2.95333111962387765731102884200, 3.63193578165173078636820807813, 3.64966428357987282931273495834, 3.85266923056098866677496472841, 4.29635119377844893824061518262, 4.94151141551248101791502101468, 4.94222407077986346501069941146, 4.98978336173385086920000277892, 5.39617893512852430428358896371, 5.93698541688893253148274222630, 6.32462612315227252053908141126, 6.32578571093637117320621063779, 6.34257202001605565835173484535, 6.93943179042441085574713281106, 7.20014988059104439849118628959, 7.33810669535277704967045103559, 7.51493357631170063068452219876, 8.202431150020850179598005786573
