| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.581945231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.581945231\) |
| \(L(1)\) |
|
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
−5.79603441797551134471461948220, −5.66572547180933865418266382475, −5.25316538475444672452793224515, −5.08545380510098788763134398570, −5.00680036260821202274442290663, −4.55579621350816453745062037154, −4.45187114191349202380322327941, −4.41904281599654030439105974710, −4.39190210048110796243179546391, −4.20975306712954653646542409575, −4.04024753903629619239532926112, −3.69462614611331530779297671376, −3.64396426404474128743491285048, −3.55784347309864131745363255916, −3.20605860658676969118475498155, −3.02240061166148810060897481745, −2.86286681511850626155139481458, −2.18692501391830479460546308390, −2.14464560692733619872847310644, −1.80602455890556564030537524830, −1.68586807534902657229604269788, −1.26607415877569986536904892397, −1.23928571035429499720029283197, −1.23056767879052240405904594861, −0.870417687569341541492741129852,
0.870417687569341541492741129852, 1.23056767879052240405904594861, 1.23928571035429499720029283197, 1.26607415877569986536904892397, 1.68586807534902657229604269788, 1.80602455890556564030537524830, 2.14464560692733619872847310644, 2.18692501391830479460546308390, 2.86286681511850626155139481458, 3.02240061166148810060897481745, 3.20605860658676969118475498155, 3.55784347309864131745363255916, 3.64396426404474128743491285048, 3.69462614611331530779297671376, 4.04024753903629619239532926112, 4.20975306712954653646542409575, 4.39190210048110796243179546391, 4.41904281599654030439105974710, 4.45187114191349202380322327941, 4.55579621350816453745062037154, 5.00680036260821202274442290663, 5.08545380510098788763134398570, 5.25316538475444672452793224515, 5.66572547180933865418266382475, 5.79603441797551134471461948220
