L(s) = 1 | − 2-s + 3-s + 4-s + 3.41·5-s − 6-s + 7-s − 8-s + 9-s − 3.41·10-s − 4.57·11-s + 12-s + 4.16·13-s − 14-s + 3.41·15-s + 16-s + 7.03·17-s − 18-s − 0.908·19-s + 3.41·20-s + 21-s + 4.57·22-s − 4.31·23-s − 24-s + 6.67·25-s − 4.16·26-s + 27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.52·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.08·10-s − 1.37·11-s + 0.288·12-s + 1.15·13-s − 0.267·14-s + 0.882·15-s + 0.250·16-s + 1.70·17-s − 0.235·18-s − 0.208·19-s + 0.764·20-s + 0.218·21-s + 0.975·22-s − 0.899·23-s − 0.204·24-s + 1.33·25-s − 0.817·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.927356655\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.927356655\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
good | 5 | \( 1 - 3.41T + 5T^{2} \) |
| 11 | \( 1 + 4.57T + 11T^{2} \) |
| 13 | \( 1 - 4.16T + 13T^{2} \) |
| 17 | \( 1 - 7.03T + 17T^{2} \) |
| 19 | \( 1 + 0.908T + 19T^{2} \) |
| 23 | \( 1 + 4.31T + 23T^{2} \) |
| 29 | \( 1 + 2.45T + 29T^{2} \) |
| 31 | \( 1 - 8.12T + 31T^{2} \) |
| 37 | \( 1 - 9.34T + 37T^{2} \) |
| 41 | \( 1 - 0.641T + 41T^{2} \) |
| 43 | \( 1 + 0.555T + 43T^{2} \) |
| 47 | \( 1 - 5.14T + 47T^{2} \) |
| 53 | \( 1 + 4.44T + 53T^{2} \) |
| 59 | \( 1 + 1.64T + 59T^{2} \) |
| 61 | \( 1 - 6.38T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 - 3.74T + 71T^{2} \) |
| 73 | \( 1 - 2.36T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 3.05T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−7.989787255655775542796551350855, −7.42559792064259652188707711145, −6.32042425148213129788217113967, −5.86320534508949846952173954959, −5.31595676809538800237263434976, −4.26202082981913129019751676115, −3.12102084837585736738604207240, −2.54219500352811571952273824971, −1.73140056278332267912876050321, −0.975692570083135834314728072488,
0.975692570083135834314728072488, 1.73140056278332267912876050321, 2.54219500352811571952273824971, 3.12102084837585736738604207240, 4.26202082981913129019751676115, 5.31595676809538800237263434976, 5.86320534508949846952173954959, 6.32042425148213129788217113967, 7.42559792064259652188707711145, 7.989787255655775542796551350855