| L(s) = 1 | − 729·9-s + 4.53e3·19-s − 1.18e5·31-s + 7.76e4·49-s + 7.15e5·61-s + 4.09e5·79-s + 5.31e5·81-s − 4.61e6·109-s + 3.54e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6.61e6·169-s − 3.30e6·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 9-s + 0.661·19-s − 3.97·31-s + 0.660·49-s + 3.15·61-s + 0.830·79-s + 81-s − 3.56·109-s + 2·121-s − 1.36·169-s − 0.661·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.104592009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.104592009\) |
| \(L(4)\) |
|
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 | $C_2$ | \( 1 + p^{6} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 77689 T^{2} + p^{12} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6611471 T^{2} + p^{12} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2269 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 59221 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2826257618 T^{2} + p^{12} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 10829073529 T^{2} + p^{12} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 357839 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 161772883271 T^{2} + p^{12} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 104459767778 T^{2} + p^{12} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 204622 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 920565867311 T^{2} + p^{12} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.08853066015320111979826677246, −10.65429455092219709837284509408, −9.892976314461491930281801968425, −9.516942032956331589027753922539, −8.996752608758585603274304436582, −8.708556068011662620605278705985, −8.060241346871258143093217864640, −7.60194793342104024179867135268, −7.05406940444134086384470866148, −6.67724390285902828135259159212, −5.81989664002203358651066969135, −5.35507045077673847229130685196, −5.30909218480297973879525105757, −4.22531624188475232738595068112, −3.64157157541717725538262580163, −3.27110003254494173381221212973, −2.37664783184488183604618671941, −1.94476470346742951205703976085, −1.05272140080950736050339587729, −0.27676132728551748362873993274,
0.27676132728551748362873993274, 1.05272140080950736050339587729, 1.94476470346742951205703976085, 2.37664783184488183604618671941, 3.27110003254494173381221212973, 3.64157157541717725538262580163, 4.22531624188475232738595068112, 5.30909218480297973879525105757, 5.35507045077673847229130685196, 5.81989664002203358651066969135, 6.67724390285902828135259159212, 7.05406940444134086384470866148, 7.60194793342104024179867135268, 8.060241346871258143093217864640, 8.708556068011662620605278705985, 8.996752608758585603274304436582, 9.516942032956331589027753922539, 9.892976314461491930281801968425, 10.65429455092219709837284509408, 11.08853066015320111979826677246
