| L(s) = 1 | + 1.67·2-s + 0.795·4-s + 1.45·5-s − 0.113·7-s − 2.01·8-s + 2.44·10-s + 11-s − 2.41·13-s − 0.190·14-s − 4.95·16-s − 4.33·17-s + 5.52·19-s + 1.16·20-s + 1.67·22-s + 5.61·23-s − 2.86·25-s − 4.04·26-s − 0.0906·28-s + 2.97·29-s + 3.23·31-s − 4.26·32-s − 7.25·34-s − 0.166·35-s + 5.47·37-s + 9.24·38-s − 2.93·40-s + 3.54·41-s + ⋯ |
| L(s) = 1 | + 1.18·2-s + 0.397·4-s + 0.652·5-s − 0.0430·7-s − 0.711·8-s + 0.771·10-s + 0.301·11-s − 0.670·13-s − 0.0509·14-s − 1.23·16-s − 1.05·17-s + 1.26·19-s + 0.259·20-s + 0.356·22-s + 1.17·23-s − 0.573·25-s − 0.793·26-s − 0.0171·28-s + 0.552·29-s + 0.581·31-s − 0.753·32-s − 1.24·34-s − 0.0281·35-s + 0.900·37-s + 1.49·38-s − 0.464·40-s + 0.554·41-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
| \(L(1)\) |
\(\approx\) |
\(3.747677878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.747677878\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.67T + 2T^{2} \) |
| 5 | \( 1 - 1.45T + 5T^{2} \) |
| 7 | \( 1 + 0.113T + 7T^{2} \) |
| 13 | \( 1 + 2.41T + 13T^{2} \) |
| 17 | \( 1 + 4.33T + 17T^{2} \) |
| 19 | \( 1 - 5.52T + 19T^{2} \) |
| 23 | \( 1 - 5.61T + 23T^{2} \) |
| 29 | \( 1 - 2.97T + 29T^{2} \) |
| 31 | \( 1 - 3.23T + 31T^{2} \) |
| 37 | \( 1 - 5.47T + 37T^{2} \) |
| 41 | \( 1 - 3.54T + 41T^{2} \) |
| 43 | \( 1 - 3.64T + 43T^{2} \) |
| 47 | \( 1 - 6.81T + 47T^{2} \) |
| 53 | \( 1 - 0.123T + 53T^{2} \) |
| 59 | \( 1 - 14.8T + 59T^{2} \) |
| 67 | \( 1 - 9.16T + 67T^{2} \) |
| 71 | \( 1 - 9.69T + 71T^{2} \) |
| 73 | \( 1 + 1.05T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 2.01T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 4.43T + 97T^{2} \) |
| 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
−8.004635683792721137245867854146, −7.02092520641100181588265698062, −6.55313351107885725593034692754, −5.72639199233544018766250380374, −5.18445126302614715099943882889, −4.52073052460563467982330655852, −3.78276155701650674549205591884, −2.82266899163281572795023265365, −2.27155712664817350507168110389, −0.849488502495908868069502440206,
0.849488502495908868069502440206, 2.27155712664817350507168110389, 2.82266899163281572795023265365, 3.78276155701650674549205591884, 4.52073052460563467982330655852, 5.18445126302614715099943882889, 5.72639199233544018766250380374, 6.55313351107885725593034692754, 7.02092520641100181588265698062, 8.004635683792721137245867854146
