| L(s) = 1 | + 2·9-s − 4·11-s + 4·43-s − 4·67-s + 81-s + 4·83-s − 4·97-s − 8·99-s + 8·121-s + 127-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 + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 2·9-s − 4·11-s + 4·43-s − 4·67-s + 81-s + 4·83-s − 4·97-s − 8·99-s + 8·121-s + 127-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 + 211-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9347629425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9347629425\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 5 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 7 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
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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
−7.02414730682565773663169093481, −6.71406947790972227507533447445, −6.42934681294388666830456057040, −6.13191206271410916017790363166, −6.08147011726916074779471198215, −5.58879484636940979753433714076, −5.58864208991676926467501659130, −5.46343482179326755314663505136, −5.21047905721699790842717326679, −4.83984312061834445814608165338, −4.64668023116325041777032108756, −4.59491423057164344259054663053, −4.14125378513960999213989410835, −4.12546933301071398555441832496, −3.99032780927207764418018475307, −3.35266426372405733648033800729, −3.11634578268721646686176094240, −2.97646602586126980552600761433, −2.66344206676534705024145389274, −2.51670679715179801950478913116, −2.14855686489585693498138425821, −1.90950780038349182791951989819, −1.58274666754041462145847937560, −1.06619462788161135981329501116, −0.59389852830116037329931522557,
0.59389852830116037329931522557, 1.06619462788161135981329501116, 1.58274666754041462145847937560, 1.90950780038349182791951989819, 2.14855686489585693498138425821, 2.51670679715179801950478913116, 2.66344206676534705024145389274, 2.97646602586126980552600761433, 3.11634578268721646686176094240, 3.35266426372405733648033800729, 3.99032780927207764418018475307, 4.12546933301071398555441832496, 4.14125378513960999213989410835, 4.59491423057164344259054663053, 4.64668023116325041777032108756, 4.83984312061834445814608165338, 5.21047905721699790842717326679, 5.46343482179326755314663505136, 5.58864208991676926467501659130, 5.58879484636940979753433714076, 6.08147011726916074779471198215, 6.13191206271410916017790363166, 6.42934681294388666830456057040, 6.71406947790972227507533447445, 7.02414730682565773663169093481
