L(s) = 1 | + 3·5-s − 8-s − 3·13-s − 3·19-s + 6·25-s − 27-s − 3·40-s − 9·65-s + 3·67-s − 3·71-s − 9·95-s + 3·104-s + 3·121-s + 10·125-s + 127-s + 131-s − 3·135-s + 137-s + 139-s + 149-s + 151-s + 3·152-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + ⋯ |
L(s) = 1 | + 3·5-s − 8-s − 3·13-s − 3·19-s + 6·25-s − 27-s − 3·40-s − 9·65-s + 3·67-s − 3·71-s − 9·95-s + 3·104-s + 3·121-s + 10·125-s + 127-s + 131-s − 3·135-s + 137-s + 139-s + 149-s + 151-s + 3·152-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37595375 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37595375 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6413277561\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6413277561\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 67 | $C_1$ | \( ( 1 - T )^{3} \) |
good | 2 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 3 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 7 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 43 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 53 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 59 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 61 | $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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 89 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 97 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.41798551668896672858128052855, −10.22902589876060478282363114307, −9.829114752824239301839579745165, −9.726120685350341513084077659429, −9.374343001607053144156478783777, −9.149549491802112850629565265460, −8.832056223902490822940548597385, −8.352929164953226007503362542103, −8.352446113432306906431761893840, −7.43804807196807448950154101411, −7.27354473602909558849204002413, −6.92443840811092655576461129943, −6.41843497820471896040389423874, −6.30570089708396643621497282690, −5.98012143876439665013731833059, −5.64131577602335512212995357024, −5.09596984702261536648052383836, −5.06977199406534976849675565564, −4.52874915155761929951101455354, −4.15281317513195459619515169215, −3.29338205648354503099323886350, −2.59638138134959920910827783067, −2.54225804609270401666256445757, −2.10045216621666588780373750316, −1.74773988046245978161081355269,
1.74773988046245978161081355269, 2.10045216621666588780373750316, 2.54225804609270401666256445757, 2.59638138134959920910827783067, 3.29338205648354503099323886350, 4.15281317513195459619515169215, 4.52874915155761929951101455354, 5.06977199406534976849675565564, 5.09596984702261536648052383836, 5.64131577602335512212995357024, 5.98012143876439665013731833059, 6.30570089708396643621497282690, 6.41843497820471896040389423874, 6.92443840811092655576461129943, 7.27354473602909558849204002413, 7.43804807196807448950154101411, 8.352446113432306906431761893840, 8.352929164953226007503362542103, 8.832056223902490822940548597385, 9.149549491802112850629565265460, 9.374343001607053144156478783777, 9.726120685350341513084077659429, 9.829114752824239301839579745165, 10.22902589876060478282363114307, 10.41798551668896672858128052855