| L(s) = 1 | − 3-s + 3·4-s + 5-s − 3·11-s − 3·12-s − 15-s + 6·16-s + 3·20-s − 23-s + 31-s + 3·33-s + 37-s − 9·44-s + 47-s − 6·48-s + 3·49-s − 53-s − 3·55-s + 59-s − 3·60-s + 10·64-s − 6·67-s + 69-s + 71-s + 6·80-s + 89-s − 3·92-s + ⋯ |
| L(s) = 1 | − 3-s + 3·4-s + 5-s − 3·11-s − 3·12-s − 15-s + 6·16-s + 3·20-s − 23-s + 31-s + 3·33-s + 37-s − 9·44-s + 47-s − 6·48-s + 3·49-s − 53-s − 3·55-s + 59-s − 3·60-s + 10·64-s − 6·67-s + 69-s + 71-s + 6·80-s + 89-s − 3·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.883910786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.883910786\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 3 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 5 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 23 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 31 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 37 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 47 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 53 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 59 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_1$ | \( ( 1 + T )^{6} \) |
| 71 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 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 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 97 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
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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
−8.322025579780836830662388671595, −7.85243625619789117458526318481, −7.74446943293197462566540858947, −7.53782990592625448493195984670, −7.36288778847386768586868736696, −7.10073247821734180567273077415, −6.82181685127869731239612250549, −6.19101814870035695047020715233, −6.15388869568747218067232963669, −6.15202380233172224349046371869, −5.65079930615928471228446956144, −5.57609618456610737104685267280, −5.52728993749984777505149315653, −5.01268512267147233020904550420, −4.63496177075956614333509709364, −4.29704197947666420624477345383, −3.78961248791494992698185424245, −3.22677663520357625486768732187, −3.09392753285088635638293147180, −2.49959579678107293328132698038, −2.48892331493762691856556133623, −2.42820827478680535164772596065, −1.82197688265522102554459922474, −1.49616530311272384194224493215, −0.817084319002828612197594699503,
0.817084319002828612197594699503, 1.49616530311272384194224493215, 1.82197688265522102554459922474, 2.42820827478680535164772596065, 2.48892331493762691856556133623, 2.49959579678107293328132698038, 3.09392753285088635638293147180, 3.22677663520357625486768732187, 3.78961248791494992698185424245, 4.29704197947666420624477345383, 4.63496177075956614333509709364, 5.01268512267147233020904550420, 5.52728993749984777505149315653, 5.57609618456610737104685267280, 5.65079930615928471228446956144, 6.15202380233172224349046371869, 6.15388869568747218067232963669, 6.19101814870035695047020715233, 6.82181685127869731239612250549, 7.10073247821734180567273077415, 7.36288778847386768586868736696, 7.53782990592625448493195984670, 7.74446943293197462566540858947, 7.85243625619789117458526318481, 8.322025579780836830662388671595
