| L(s) = 1 | − 2-s + 2·3-s − 2·6-s − 7-s + 8-s + 3·9-s − 13-s + 14-s − 16-s − 3·18-s − 2·21-s + 2·23-s + 2·24-s + 25-s + 26-s + 4·27-s − 31-s + 37-s − 2·39-s − 41-s + 2·42-s + 2·43-s − 2·46-s − 2·48-s − 50-s − 53-s − 4·54-s + ⋯ |
| L(s) = 1 | − 2-s + 2·3-s − 2·6-s − 7-s + 8-s + 3·9-s − 13-s + 14-s − 16-s − 3·18-s − 2·21-s + 2·23-s + 2·24-s + 25-s + 26-s + 4·27-s − 31-s + 37-s − 2·39-s − 41-s + 2·42-s + 2·43-s − 2·46-s − 2·48-s − 50-s − 53-s − 4·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.239908843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.239908843\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( ( 1 - T )^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 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
−9.168016470772599494644963318248, −8.770336860632217076856473541422, −7.84317800038863663457560234242, −7.26464565512309316335020207095, −6.76146751036329385349317054842, −4.98887787415589918413267751957, −4.16979174356438955792442111940, −3.14305972073290146063425405002, −2.53324418884176460313640668536, −1.26911125277616722409566407804,
1.26911125277616722409566407804, 2.53324418884176460313640668536, 3.14305972073290146063425405002, 4.16979174356438955792442111940, 4.98887787415589918413267751957, 6.76146751036329385349317054842, 7.26464565512309316335020207095, 7.84317800038863663457560234242, 8.770336860632217076856473541422, 9.168016470772599494644963318248
