| L(s) = 1 | − 3·17-s − 27-s + 3·59-s − 3·71-s + 3·121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 3·17-s − 27-s + 3·59-s − 3·71-s + 3·121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13144256 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13144256 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3695927470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3695927470\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 59 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 5 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 7 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 19 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 41 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 53 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 79 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.39528022992789676930590622121, −10.73185662055696916936436795585, −10.58511910677678740321369774917, −10.15017851424962438163639135320, −9.898616490135496005271700954016, −9.433044865027425197776529546453, −9.015654839054061982886079284010, −8.959875841478556351899660191887, −8.415510902281815381397559731367, −8.373024431415770941498906861611, −7.77477370454637462466025153317, −7.21343563881083062875961979333, −7.14066791325258048277050639966, −6.78350264068296126701825459845, −6.15801222699119574672525207473, −6.14598647640397162280831672792, −5.51842220931059345605825926688, −5.12245575341788368696274205789, −4.46360128568295768700341822966, −4.46311107071320256859512756467, −3.83758519256611644826218414016, −3.43989334649824293273873876565, −2.49628236703952269598523859769, −2.39794755212876959589973761562, −1.62114114216118457721131186349,
1.62114114216118457721131186349, 2.39794755212876959589973761562, 2.49628236703952269598523859769, 3.43989334649824293273873876565, 3.83758519256611644826218414016, 4.46311107071320256859512756467, 4.46360128568295768700341822966, 5.12245575341788368696274205789, 5.51842220931059345605825926688, 6.14598647640397162280831672792, 6.15801222699119574672525207473, 6.78350264068296126701825459845, 7.14066791325258048277050639966, 7.21343563881083062875961979333, 7.77477370454637462466025153317, 8.373024431415770941498906861611, 8.415510902281815381397559731367, 8.959875841478556351899660191887, 9.015654839054061982886079284010, 9.433044865027425197776529546453, 9.898616490135496005271700954016, 10.15017851424962438163639135320, 10.58511910677678740321369774917, 10.73185662055696916936436795585, 11.39528022992789676930590622121
