L(s) = 1 | − 1.36·2-s − 0.153·3-s − 0.146·4-s − 5-s + 0.209·6-s − 7-s + 2.92·8-s − 2.97·9-s + 1.36·10-s + 0.539·11-s + 0.0225·12-s − 6.23·13-s + 1.36·14-s + 0.153·15-s − 3.68·16-s + 5.23·17-s + 4.05·18-s − 6.12·19-s + 0.146·20-s + 0.153·21-s − 0.734·22-s − 0.273·23-s − 0.449·24-s + 25-s + 8.49·26-s + 0.919·27-s + 0.146·28-s + ⋯ |
L(s) = 1 | − 0.962·2-s − 0.0888·3-s − 0.0732·4-s − 0.447·5-s + 0.0855·6-s − 0.377·7-s + 1.03·8-s − 0.992·9-s + 0.430·10-s + 0.162·11-s + 0.00650·12-s − 1.73·13-s + 0.363·14-s + 0.0397·15-s − 0.921·16-s + 1.26·17-s + 0.955·18-s − 1.40·19-s + 0.0327·20-s + 0.0335·21-s − 0.156·22-s − 0.0570·23-s − 0.0917·24-s + 0.200·25-s + 1.66·26-s + 0.176·27-s + 0.0276·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
good | 2 | \( 1 + 1.36T + 2T^{2} \) |
| 3 | \( 1 + 0.153T + 3T^{2} \) |
| 11 | \( 1 - 0.539T + 11T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 + 6.12T + 19T^{2} \) |
| 23 | \( 1 + 0.273T + 23T^{2} \) |
| 29 | \( 1 + 4.67T + 29T^{2} \) |
| 31 | \( 1 - 5.78T + 31T^{2} \) |
| 37 | \( 1 - 7.61T + 37T^{2} \) |
| 41 | \( 1 + 2.28T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 4.38T + 47T^{2} \) |
| 53 | \( 1 - 2.31T + 53T^{2} \) |
| 59 | \( 1 + 1.21T + 59T^{2} \) |
| 61 | \( 1 - 2.25T + 61T^{2} \) |
| 67 | \( 1 + 0.417T + 67T^{2} \) |
| 71 | \( 1 - 0.240T + 71T^{2} \) |
| 73 | \( 1 + 7.01T + 73T^{2} \) |
| 79 | \( 1 - 7.37T + 79T^{2} \) |
| 83 | \( 1 + 9.96T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 6.28T + 97T^{2} \) |
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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
−7.66195478963918524318124911005, −7.10923391982672221991922146062, −6.15190093911764073754816607715, −5.45133865253151266311987898516, −4.60371403530252623433732335558, −3.99768080744811993540540453495, −2.89120473232789970435123733231, −2.20712501088434028998173898324, −0.830412187607036453782625331030, 0,
0.830412187607036453782625331030, 2.20712501088434028998173898324, 2.89120473232789970435123733231, 3.99768080744811993540540453495, 4.60371403530252623433732335558, 5.45133865253151266311987898516, 6.15190093911764073754816607715, 7.10923391982672221991922146062, 7.66195478963918524318124911005