| L(s) = 1 | − 0.926·2-s − 1.14·4-s − 3.01·5-s + 2.23·7-s + 2.91·8-s + 2.79·10-s − 11-s − 3.86·13-s − 2.07·14-s − 0.411·16-s + 0.440·17-s + 3.02·19-s + 3.44·20-s + 0.926·22-s − 5.93·23-s + 4.07·25-s + 3.57·26-s − 2.55·28-s + 8.56·29-s − 0.629·31-s − 5.43·32-s − 0.408·34-s − 6.74·35-s + 5.92·37-s − 2.79·38-s − 8.76·40-s − 9.04·41-s + ⋯ |
| L(s) = 1 | − 0.654·2-s − 0.571·4-s − 1.34·5-s + 0.846·7-s + 1.02·8-s + 0.882·10-s − 0.301·11-s − 1.07·13-s − 0.554·14-s − 0.102·16-s + 0.106·17-s + 0.693·19-s + 0.769·20-s + 0.197·22-s − 1.23·23-s + 0.815·25-s + 0.701·26-s − 0.483·28-s + 1.58·29-s − 0.113·31-s − 0.961·32-s − 0.0700·34-s − 1.14·35-s + 0.974·37-s − 0.453·38-s − 1.38·40-s − 1.41·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 0.926T + 2T^{2} \) |
| 5 | \( 1 + 3.01T + 5T^{2} \) |
| 7 | \( 1 - 2.23T + 7T^{2} \) |
| 13 | \( 1 + 3.86T + 13T^{2} \) |
| 17 | \( 1 - 0.440T + 17T^{2} \) |
| 19 | \( 1 - 3.02T + 19T^{2} \) |
| 23 | \( 1 + 5.93T + 23T^{2} \) |
| 29 | \( 1 - 8.56T + 29T^{2} \) |
| 31 | \( 1 + 0.629T + 31T^{2} \) |
| 37 | \( 1 - 5.92T + 37T^{2} \) |
| 41 | \( 1 + 9.04T + 41T^{2} \) |
| 43 | \( 1 - 3.19T + 43T^{2} \) |
| 47 | \( 1 - 2.54T + 47T^{2} \) |
| 53 | \( 1 + 8.76T + 53T^{2} \) |
| 59 | \( 1 - 7.36T + 59T^{2} \) |
| 67 | \( 1 - 0.878T + 67T^{2} \) |
| 71 | \( 1 + 5.06T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 7.46T + 83T^{2} \) |
| 89 | \( 1 - 1.43T + 89T^{2} \) |
| 97 | \( 1 + 4.10T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.85274874792828695445267046035, −7.48767397931030080307639416902, −6.52450137506626893553690857299, −5.27249601631230962445065238851, −4.76147133737710203957437548105, −4.18218618463694408908872604382, −3.32910950623835917315587799465, −2.17879093456967084618765428340, −0.981264486793941781317458377021, 0,
0.981264486793941781317458377021, 2.17879093456967084618765428340, 3.32910950623835917315587799465, 4.18218618463694408908872604382, 4.76147133737710203957437548105, 5.27249601631230962445065238851, 6.52450137506626893553690857299, 7.48767397931030080307639416902, 7.85274874792828695445267046035
