L(s) = 1 | + 1.36·2-s − 0.257·3-s − 0.148·4-s − 5-s − 0.349·6-s − 7-s − 2.92·8-s − 2.93·9-s − 1.36·10-s + 2.45·11-s + 0.0381·12-s − 6.54·13-s − 1.36·14-s + 0.257·15-s − 3.68·16-s + 2.41·17-s − 3.99·18-s − 1.82·19-s + 0.148·20-s + 0.257·21-s + 3.33·22-s − 3.40·23-s + 0.751·24-s + 25-s − 8.90·26-s + 1.52·27-s + 0.148·28-s + ⋯ |
L(s) = 1 | + 0.962·2-s − 0.148·3-s − 0.0741·4-s − 0.447·5-s − 0.142·6-s − 0.377·7-s − 1.03·8-s − 0.977·9-s − 0.430·10-s + 0.739·11-s + 0.0110·12-s − 1.81·13-s − 0.363·14-s + 0.0663·15-s − 0.920·16-s + 0.585·17-s − 0.940·18-s − 0.419·19-s + 0.0331·20-s + 0.0561·21-s + 0.711·22-s − 0.709·23-s + 0.153·24-s + 0.200·25-s − 1.74·26-s + 0.293·27-s + 0.0280·28-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
\(L(1)\) |
\(\approx\) |
\(0.8126837733\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8126837733\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.257T + 3T^{2} \) |
| 11 | \( 1 - 2.45T + 11T^{2} \) |
| 13 | \( 1 + 6.54T + 13T^{2} \) |
| 17 | \( 1 - 2.41T + 17T^{2} \) |
| 19 | \( 1 + 1.82T + 19T^{2} \) |
| 23 | \( 1 + 3.40T + 23T^{2} \) |
| 29 | \( 1 + 6.44T + 29T^{2} \) |
| 31 | \( 1 + 5.10T + 31T^{2} \) |
| 37 | \( 1 + 4.22T + 37T^{2} \) |
| 41 | \( 1 - 2.29T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 - 2.83T + 53T^{2} \) |
| 59 | \( 1 + 2.81T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 + 6.06T + 67T^{2} \) |
| 71 | \( 1 - 5.47T + 71T^{2} \) |
| 73 | \( 1 - 4.63T + 73T^{2} \) |
| 79 | \( 1 + 0.283T + 79T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−7.75458686802721874579638333215, −6.98284797752945281776622131178, −6.32088337990853030723842293200, −5.47194584828473828774753216803, −5.18537500072603811336144367962, −4.19690409232893156971135345500, −3.68120259483118043363184524868, −2.91176953050943736601067836510, −2.09585891272330349586040579338, −0.36226164702626314854019092635,
0.36226164702626314854019092635, 2.09585891272330349586040579338, 2.91176953050943736601067836510, 3.68120259483118043363184524868, 4.19690409232893156971135345500, 5.18537500072603811336144367962, 5.47194584828473828774753216803, 6.32088337990853030723842293200, 6.98284797752945281776622131178, 7.75458686802721874579638333215