| L(s) = 1 | + 2-s + 7-s + 5·9-s − 5·11-s + 14-s + 17-s + 5·18-s + 19-s − 5·22-s − 23-s + 5·25-s − 31-s + 34-s − 37-s + 38-s − 46-s − 47-s + 5·50-s + 61-s − 62-s + 5·63-s − 67-s − 74-s − 5·77-s + 79-s + 15·81-s − 89-s + ⋯ |
| L(s) = 1 | + 2-s + 7-s + 5·9-s − 5·11-s + 14-s + 17-s + 5·18-s + 19-s − 5·22-s − 23-s + 5·25-s − 31-s + 34-s − 37-s + 38-s − 46-s − 47-s + 5·50-s + 61-s − 62-s + 5·63-s − 67-s − 74-s − 5·77-s + 79-s + 15·81-s − 89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{5} \cdot 173^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{5} \cdot 173^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.632059305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.632059305\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{5} \) |
| 173 | $C_1$ | \( ( 1 + T )^{5} \) |
| good | 2 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 7 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 17 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 19 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 23 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 31 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 37 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 47 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 61 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 67 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 79 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 89 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.37865930687135433385055100638, −5.26350833224326264147746998765, −5.25552159118250605587484461893, −5.15747560978213362666225046646, −5.14240602662122057343611882206, −4.77927935546528018192954274388, −4.65926995240854212616357403041, −4.51363741641435636923881899845, −4.49764244519647889289010825392, −4.43231566366994433338703148181, −4.02140412876215456032878766399, −3.74993044428763146516552072390, −3.58650983386390565016676229650, −3.37184694506354828662404726948, −3.17423458350501447371143722398, −3.02856984884119464607206628178, −2.77913629530442497677583861897, −2.57814267861748649223700590418, −2.21040656269944498863822735305, −1.95599031973881560612950283268, −1.94975562803056593848756991874, −1.68156733437514555795934057140, −1.14175180763128206113260230746, −1.01624721225229246764366344875, −0.953264979476204804703556015653,
0.953264979476204804703556015653, 1.01624721225229246764366344875, 1.14175180763128206113260230746, 1.68156733437514555795934057140, 1.94975562803056593848756991874, 1.95599031973881560612950283268, 2.21040656269944498863822735305, 2.57814267861748649223700590418, 2.77913629530442497677583861897, 3.02856984884119464607206628178, 3.17423458350501447371143722398, 3.37184694506354828662404726948, 3.58650983386390565016676229650, 3.74993044428763146516552072390, 4.02140412876215456032878766399, 4.43231566366994433338703148181, 4.49764244519647889289010825392, 4.51363741641435636923881899845, 4.65926995240854212616357403041, 4.77927935546528018192954274388, 5.14240602662122057343611882206, 5.15747560978213362666225046646, 5.25552159118250605587484461893, 5.26350833224326264147746998765, 5.37865930687135433385055100638
