| L(s) = 1 | + 2·2-s + 3·4-s + 7-s + 4·8-s − 9-s − 11-s + 2·14-s + 5·16-s − 2·18-s − 2·22-s − 2·23-s − 25-s + 3·28-s − 2·29-s + 6·32-s − 3·36-s − 43-s − 3·44-s − 4·46-s + 49-s − 2·50-s + 2·53-s + 4·56-s − 4·58-s − 63-s + 7·64-s + 2·67-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s + 7-s + 4·8-s − 9-s − 11-s + 2·14-s + 5·16-s − 2·18-s − 2·22-s − 2·23-s − 25-s + 3·28-s − 2·29-s + 6·32-s − 3·36-s − 43-s − 3·44-s − 4·46-s + 49-s − 2·50-s + 2·53-s + 4·56-s − 4·58-s − 63-s + 7·64-s + 2·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.436512507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.436512507\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( ( 1 - T )^{2} \) |
| 3 | \( 1 + T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 + T )^{2} \) |
| 29 | \( ( 1 + T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−8.327529048966035595447667808121, −7.86576770894213139238402564395, −7.21276800265659617620724937401, −6.10546057474397191428028034742, −5.55683797397808236356124878961, −5.16058185068508082676236724432, −4.12271596725253729989644227068, −3.56122613556173330407350824039, −2.37195125549253654885749496071, −1.94810746962855116347226891179,
1.94810746962855116347226891179, 2.37195125549253654885749496071, 3.56122613556173330407350824039, 4.12271596725253729989644227068, 5.16058185068508082676236724432, 5.55683797397808236356124878961, 6.10546057474397191428028034742, 7.21276800265659617620724937401, 7.86576770894213139238402564395, 8.327529048966035595447667808121
