L(s) = 1 | − 2-s + 0.0519·3-s + 4-s − 2.57·5-s − 0.0519·6-s + 3.17·7-s − 8-s − 2.99·9-s + 2.57·10-s + 1.21·11-s + 0.0519·12-s − 6.52·13-s − 3.17·14-s − 0.133·15-s + 16-s + 0.682·17-s + 2.99·18-s + 2.63·19-s − 2.57·20-s + 0.164·21-s − 1.21·22-s + 2.04·23-s − 0.0519·24-s + 1.64·25-s + 6.52·26-s − 0.311·27-s + 3.17·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.0299·3-s + 0.5·4-s − 1.15·5-s − 0.0211·6-s + 1.19·7-s − 0.353·8-s − 0.999·9-s + 0.815·10-s + 0.367·11-s + 0.0149·12-s − 1.81·13-s − 0.848·14-s − 0.0345·15-s + 0.250·16-s + 0.165·17-s + 0.706·18-s + 0.604·19-s − 0.576·20-s + 0.0359·21-s − 0.259·22-s + 0.427·23-s − 0.0105·24-s + 0.329·25-s + 1.27·26-s − 0.0599·27-s + 0.599·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
good | 3 | \( 1 - 0.0519T + 3T^{2} \) |
| 5 | \( 1 + 2.57T + 5T^{2} \) |
| 7 | \( 1 - 3.17T + 7T^{2} \) |
| 11 | \( 1 - 1.21T + 11T^{2} \) |
| 13 | \( 1 + 6.52T + 13T^{2} \) |
| 17 | \( 1 - 0.682T + 17T^{2} \) |
| 19 | \( 1 - 2.63T + 19T^{2} \) |
| 23 | \( 1 - 2.04T + 23T^{2} \) |
| 29 | \( 1 - 7.83T + 29T^{2} \) |
| 31 | \( 1 - 7.65T + 31T^{2} \) |
| 37 | \( 1 - 0.366T + 37T^{2} \) |
| 41 | \( 1 - 1.87T + 41T^{2} \) |
| 43 | \( 1 + 3.14T + 43T^{2} \) |
| 47 | \( 1 - 1.74T + 47T^{2} \) |
| 53 | \( 1 + 0.658T + 53T^{2} \) |
| 59 | \( 1 - 1.65T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 2.72T + 71T^{2} \) |
| 73 | \( 1 + 2.31T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 2.41T + 83T^{2} \) |
| 89 | \( 1 + 0.424T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−8.118346691450566623417056557201, −7.55765211300702463885414824851, −6.96311192210065407328241358435, −5.89935656598324685249497532480, −4.88648059652378841879532217303, −4.47617166899807359366722092926, −3.15242598214257044061320551187, −2.50963327576044893581693972008, −1.19406508825594089487485946181, 0,
1.19406508825594089487485946181, 2.50963327576044893581693972008, 3.15242598214257044061320551187, 4.47617166899807359366722092926, 4.88648059652378841879532217303, 5.89935656598324685249497532480, 6.96311192210065407328241358435, 7.55765211300702463885414824851, 8.118346691450566623417056557201