L(s) = 1 | − 2.71·2-s − 3-s + 5.38·4-s − 0.137·5-s + 2.71·6-s − 9.20·8-s + 9-s + 0.374·10-s + 5.69·11-s − 5.38·12-s − 2.94·13-s + 0.137·15-s + 14.2·16-s − 2.25·17-s − 2.71·18-s + 2.05·19-s − 0.742·20-s − 15.4·22-s + 8.70·23-s + 9.20·24-s − 4.98·25-s + 8.01·26-s − 27-s + 1.76·29-s − 0.374·30-s − 5.49·31-s − 20.2·32-s + ⋯ |
L(s) = 1 | − 1.92·2-s − 0.577·3-s + 2.69·4-s − 0.0616·5-s + 1.10·6-s − 3.25·8-s + 0.333·9-s + 0.118·10-s + 1.71·11-s − 1.55·12-s − 0.817·13-s + 0.0356·15-s + 3.56·16-s − 0.547·17-s − 0.640·18-s + 0.470·19-s − 0.166·20-s − 3.30·22-s + 1.81·23-s + 1.87·24-s − 0.996·25-s + 1.57·26-s − 0.192·27-s + 0.326·29-s − 0.0684·30-s − 0.986·31-s − 3.58·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 + 2.71T + 2T^{2} \) |
| 5 | \( 1 + 0.137T + 5T^{2} \) |
| 11 | \( 1 - 5.69T + 11T^{2} \) |
| 13 | \( 1 + 2.94T + 13T^{2} \) |
| 17 | \( 1 + 2.25T + 17T^{2} \) |
| 19 | \( 1 - 2.05T + 19T^{2} \) |
| 23 | \( 1 - 8.70T + 23T^{2} \) |
| 29 | \( 1 - 1.76T + 29T^{2} \) |
| 31 | \( 1 + 5.49T + 31T^{2} \) |
| 37 | \( 1 + 4.72T + 37T^{2} \) |
| 43 | \( 1 + 1.02T + 43T^{2} \) |
| 47 | \( 1 + 8.56T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 14.6T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 3.37T + 67T^{2} \) |
| 71 | \( 1 + 5.43T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 8.16T + 79T^{2} \) |
| 83 | \( 1 - 7.37T + 83T^{2} \) |
| 89 | \( 1 - 0.423T + 89T^{2} \) |
| 97 | \( 1 + 5.89T + 97T^{2} \) |
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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
−7.65593895693099021339892170002, −7.10550919975541450775573190652, −6.70231230424203898164890280158, −5.96289395071790888602150356002, −5.03759946846066718189642261037, −3.86723766414560342533740144807, −2.88499961393531636705660155137, −1.79769256898575243297076277080, −1.12154401593767170485453607918, 0,
1.12154401593767170485453607918, 1.79769256898575243297076277080, 2.88499961393531636705660155137, 3.86723766414560342533740144807, 5.03759946846066718189642261037, 5.96289395071790888602150356002, 6.70231230424203898164890280158, 7.10550919975541450775573190652, 7.65593895693099021339892170002