| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 10-s + 11-s + 12-s − 13-s − 15-s + 16-s − 2·19-s − 20-s − 22-s − 24-s + 26-s − 27-s + 29-s + 30-s + 31-s − 32-s + 33-s + 2·38-s − 39-s + 40-s + 43-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 10-s + 11-s + 12-s − 13-s − 15-s + 16-s − 2·19-s − 20-s − 22-s − 24-s + 26-s − 27-s + 29-s + 30-s + 31-s − 32-s + 33-s + 2·38-s − 39-s + 40-s + 43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4610161263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4610161263\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - T + T^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 + T )^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−14.16016390120846201257187485941, −12.46029134035790053492481655909, −11.68925523873542102048724869802, −10.51454380665689720515485244983, −9.266948877644499009961692823650, −8.471244424587930683412558479394, −7.66013121636265313822893246199, −6.46520354613878822194579118896, −4.06320898332710260335075027351, −2.51709295006250415119800299093,
2.51709295006250415119800299093, 4.06320898332710260335075027351, 6.46520354613878822194579118896, 7.66013121636265313822893246199, 8.471244424587930683412558479394, 9.266948877644499009961692823650, 10.51454380665689720515485244983, 11.68925523873542102048724869802, 12.46029134035790053492481655909, 14.16016390120846201257187485941
