L(s) = 1 | − 2.23·2-s + 1.32·3-s + 3.01·4-s + 3.90·5-s − 2.96·6-s − 1.03·7-s − 2.26·8-s − 1.24·9-s − 8.74·10-s − 3.76·11-s + 3.99·12-s + 13-s + 2.31·14-s + 5.18·15-s − 0.956·16-s − 1.19·17-s + 2.78·18-s + 5.93·19-s + 11.7·20-s − 1.36·21-s + 8.43·22-s + 5.47·23-s − 2.99·24-s + 10.2·25-s − 2.23·26-s − 5.62·27-s − 3.10·28-s + ⋯ |
L(s) = 1 | − 1.58·2-s + 0.765·3-s + 1.50·4-s + 1.74·5-s − 1.21·6-s − 0.390·7-s − 0.799·8-s − 0.414·9-s − 2.76·10-s − 1.13·11-s + 1.15·12-s + 0.277·13-s + 0.617·14-s + 1.33·15-s − 0.239·16-s − 0.289·17-s + 0.655·18-s + 1.36·19-s + 2.63·20-s − 0.298·21-s + 1.79·22-s + 1.14·23-s − 0.612·24-s + 2.05·25-s − 0.439·26-s − 1.08·27-s − 0.587·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 619 | \( 1 - T \) |
good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 - 1.32T + 3T^{2} \) |
| 5 | \( 1 - 3.90T + 5T^{2} \) |
| 7 | \( 1 + 1.03T + 7T^{2} \) |
| 11 | \( 1 + 3.76T + 11T^{2} \) |
| 17 | \( 1 + 1.19T + 17T^{2} \) |
| 19 | \( 1 - 5.93T + 19T^{2} \) |
| 23 | \( 1 - 5.47T + 23T^{2} \) |
| 29 | \( 1 + 8.46T + 29T^{2} \) |
| 31 | \( 1 + 6.03T + 31T^{2} \) |
| 37 | \( 1 - 4.09T + 37T^{2} \) |
| 41 | \( 1 + 5.95T + 41T^{2} \) |
| 43 | \( 1 + 0.0499T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 2.35T + 53T^{2} \) |
| 59 | \( 1 + 2.92T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 - 2.07T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 + 1.14T + 89T^{2} \) |
| 97 | \( 1 - 0.332T + 97T^{2} \) |
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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
−7.62413862436561463425461944389, −7.10530144678947399421719331198, −6.25659273383378258166159257746, −5.55037506993689655902451765588, −4.99561346308656269936188014159, −3.32626282212119214753894141972, −2.78464826600979559490598740629, −2.01705900947730805288221850675, −1.38253800878344227466815929422, 0,
1.38253800878344227466815929422, 2.01705900947730805288221850675, 2.78464826600979559490598740629, 3.32626282212119214753894141972, 4.99561346308656269936188014159, 5.55037506993689655902451765588, 6.25659273383378258166159257746, 7.10530144678947399421719331198, 7.62413862436561463425461944389