| L(s) = 1 | − 8·4-s + 5·9-s + 40·16-s − 20·31-s − 40·36-s + 28·49-s − 160·64-s + 16·81-s − 22·121-s + 160·124-s + 127-s + 131-s + 137-s + 139-s + 200·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 224·196-s + ⋯ |
| L(s) = 1 | − 4·4-s + 5/3·9-s + 10·16-s − 3.59·31-s − 6.66·36-s + 4·49-s − 20·64-s + 16/9·81-s − 2·121-s + 14.3·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 50/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 16·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6447436680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6447436680\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2}( 1 + 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2}( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 95 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.43181015916288493113809601077, −7.27880225648591179336223759098, −6.97113932821447455354628354830, −6.66616626958352135965378349550, −6.50782503277666358392566002653, −5.85178239071061506736739649709, −5.69128670001264019738298771088, −5.61078433429322471167381449637, −5.39624003169409678417574263387, −5.35031418448083963065659489179, −4.76763387295287100738563281793, −4.75782028154387791004990478000, −4.41297941920619156399297024038, −4.18076235095347934769139061450, −4.07750982349987772190396743637, −3.80737018210525945412835870765, −3.61731115066062440854324804012, −3.39934128202570735807086305435, −3.05999526166417652689461231315, −2.50011227736513341061658032683, −1.96420001903979055192800385570, −1.60823952968475727898726893515, −1.18082830095439656368622583428, −0.792718094811562857389913547233, −0.33467337291877393707617747332,
0.33467337291877393707617747332, 0.792718094811562857389913547233, 1.18082830095439656368622583428, 1.60823952968475727898726893515, 1.96420001903979055192800385570, 2.50011227736513341061658032683, 3.05999526166417652689461231315, 3.39934128202570735807086305435, 3.61731115066062440854324804012, 3.80737018210525945412835870765, 4.07750982349987772190396743637, 4.18076235095347934769139061450, 4.41297941920619156399297024038, 4.75782028154387791004990478000, 4.76763387295287100738563281793, 5.35031418448083963065659489179, 5.39624003169409678417574263387, 5.61078433429322471167381449637, 5.69128670001264019738298771088, 5.85178239071061506736739649709, 6.50782503277666358392566002653, 6.66616626958352135965378349550, 6.97113932821447455354628354830, 7.27880225648591179336223759098, 7.43181015916288493113809601077
