| L(s) = 1 | − 2-s + 3-s − 5-s − 6-s + 8-s + 9-s + 10-s − 15-s − 16-s + 2·17-s − 18-s − 19-s + 24-s + 27-s + 30-s − 31-s − 2·34-s + 2·37-s + 38-s − 40-s + 2·41-s + 2·43-s − 45-s − 48-s + 49-s + 2·51-s − 53-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s − 5-s − 6-s + 8-s + 9-s + 10-s − 15-s − 16-s + 2·17-s − 18-s − 19-s + 24-s + 27-s + 30-s − 31-s − 2·34-s + 2·37-s + 38-s − 40-s + 2·41-s + 2·43-s − 45-s − 48-s + 49-s + 2·51-s − 53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7376993348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7376993348\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 397 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )^{2} \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.616922658561449429930030547494, −9.194080004641926751036044273253, −8.222039205283868382997735291931, −7.71739439227907536926801467811, −7.35298026581902640791811405321, −5.85487226162780725774481415817, −4.39250056628993093444390056108, −3.87651932799014602194853317059, −2.63011505841509572279922587200, −1.15050973480372169341303147778,
1.15050973480372169341303147778, 2.63011505841509572279922587200, 3.87651932799014602194853317059, 4.39250056628993093444390056108, 5.85487226162780725774481415817, 7.35298026581902640791811405321, 7.71739439227907536926801467811, 8.222039205283868382997735291931, 9.194080004641926751036044273253, 9.616922658561449429930030547494
