| L(s) = 1 | − 2·2-s + 4-s + 2·8-s − 4·16-s − 8·29-s + 2·32-s + 16·58-s + 3·64-s − 8·116-s − 2·121-s + 127-s − 6·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2·2-s + 4-s + 2·8-s − 4·16-s − 8·29-s + 2·32-s + 16·58-s + 3·64-s − 8·116-s − 2·121-s + 127-s − 6·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1154271626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1154271626\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_1$ | \( ( 1 + T )^{8} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
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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
−7.10366130523002705997653649497, −7.02815258369489530782367059805, −6.35973050890619636035788423990, −6.15735368025110331681855042333, −6.11190708894644019578999840838, −5.65910093255353624535780162829, −5.53641733325952376367023765449, −5.34344098036039832425231811150, −5.29401768510124845186256481067, −4.83802481538778707204668618847, −4.82107656251439584939117241219, −4.19436994871180207183784261692, −4.12641046610760009615627387946, −3.93631867236240465507806718378, −3.86118595625707887904445289356, −3.50391329763465637679242450302, −3.26916145146759369928842089433, −2.93786377108900925863387528571, −2.44210338167708534506680586034, −2.04660180186693198617227803882, −1.91104522520550825427703019289, −1.71010602305280053563800764311, −1.59659538639705878876861769756, −0.947307761203043917555041474569, −0.31003323001282393933179788417,
0.31003323001282393933179788417, 0.947307761203043917555041474569, 1.59659538639705878876861769756, 1.71010602305280053563800764311, 1.91104522520550825427703019289, 2.04660180186693198617227803882, 2.44210338167708534506680586034, 2.93786377108900925863387528571, 3.26916145146759369928842089433, 3.50391329763465637679242450302, 3.86118595625707887904445289356, 3.93631867236240465507806718378, 4.12641046610760009615627387946, 4.19436994871180207183784261692, 4.82107656251439584939117241219, 4.83802481538778707204668618847, 5.29401768510124845186256481067, 5.34344098036039832425231811150, 5.53641733325952376367023765449, 5.65910093255353624535780162829, 6.11190708894644019578999840838, 6.15735368025110331681855042333, 6.35973050890619636035788423990, 7.02815258369489530782367059805, 7.10366130523002705997653649497
