L(s) = 1 | + 2-s + 4-s + 1.64·5-s + 8-s + 1.64·10-s − 1.64·11-s + 0.645·13-s + 16-s − 1.64·17-s + 2·19-s + 1.64·20-s − 1.64·22-s + 9.29·23-s − 2.29·25-s + 0.645·26-s + 7.64·29-s + 0.645·31-s + 32-s − 1.64·34-s + 3.93·37-s + 2·38-s + 1.64·40-s − 4.93·41-s + 5·43-s − 1.64·44-s + 9.29·46-s − 10.9·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.736·5-s + 0.353·8-s + 0.520·10-s − 0.496·11-s + 0.179·13-s + 0.250·16-s − 0.399·17-s + 0.458·19-s + 0.368·20-s − 0.350·22-s + 1.93·23-s − 0.458·25-s + 0.126·26-s + 1.41·29-s + 0.115·31-s + 0.176·32-s − 0.282·34-s + 0.647·37-s + 0.324·38-s + 0.260·40-s − 0.771·41-s + 0.762·43-s − 0.248·44-s + 1.36·46-s − 1.59·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.431229540\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.431229540\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 1.64T + 5T^{2} \) |
| 11 | \( 1 + 1.64T + 11T^{2} \) |
| 13 | \( 1 - 0.645T + 13T^{2} \) |
| 17 | \( 1 + 1.64T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 9.29T + 23T^{2} \) |
| 29 | \( 1 - 7.64T + 29T^{2} \) |
| 31 | \( 1 - 0.645T + 31T^{2} \) |
| 37 | \( 1 - 3.93T + 37T^{2} \) |
| 41 | \( 1 + 4.93T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 - 8.29T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 + 2.70T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 7.58T + 97T^{2} \) |
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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
−8.800911124345058176243621146240, −8.116574850442499993552062840304, −7.02882458394887368259012825011, −6.58745089550047259641725770613, −5.55427470469490270124761544967, −5.10644225656728019526561751694, −4.15797439492400384816457250208, −3.05740726886877631625062705561, −2.36286529542835151260423417564, −1.11467340249133286939919662557,
1.11467340249133286939919662557, 2.36286529542835151260423417564, 3.05740726886877631625062705561, 4.15797439492400384816457250208, 5.10644225656728019526561751694, 5.55427470469490270124761544967, 6.58745089550047259641725770613, 7.02882458394887368259012825011, 8.116574850442499993552062840304, 8.800911124345058176243621146240