| L(s) = 1 | − 2-s + 7-s + 8-s − 11-s − 14-s − 16-s + 22-s + 2·23-s + 25-s + 2·29-s − 37-s − 43-s − 2·46-s + 49-s − 50-s − 53-s + 56-s − 2·58-s + 64-s − 67-s − 71-s + 74-s − 77-s − 79-s + 86-s − 88-s − 98-s + ⋯ |
| L(s) = 1 | − 2-s + 7-s + 8-s − 11-s − 14-s − 16-s + 22-s + 2·23-s + 25-s + 2·29-s − 37-s − 43-s − 2·46-s + 49-s − 50-s − 53-s + 56-s − 2·58-s + 64-s − 67-s − 71-s + 74-s − 77-s − 79-s + 86-s − 88-s − 98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5602299567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5602299567\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )^{2} \) |
| 29 | \( ( 1 - T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 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
−10.65486222975850203163492152625, −10.22463290015606000178750212343, −8.957282327605518121662427480932, −8.488344325266820694893448720970, −7.64498140303305301374911459239, −6.80515368051977052850892646099, −5.13617309222343337825309414293, −4.67049883407006982822049927611, −2.87071877068400395621172717010, −1.30083559297036396805354158382,
1.30083559297036396805354158382, 2.87071877068400395621172717010, 4.67049883407006982822049927611, 5.13617309222343337825309414293, 6.80515368051977052850892646099, 7.64498140303305301374911459239, 8.488344325266820694893448720970, 8.957282327605518121662427480932, 10.22463290015606000178750212343, 10.65486222975850203163492152625
