| L(s) = 1 | + 2.63·2-s − 3.04·3-s + 4.91·4-s − 8.00·6-s + 0.574·7-s + 7.67·8-s + 6.26·9-s + 2.57·11-s − 14.9·12-s + 0.468·13-s + 1.51·14-s + 10.3·16-s + 4.08·17-s + 16.4·18-s + 19-s − 1.74·21-s + 6.77·22-s − 1.51·23-s − 23.3·24-s + 1.23·26-s − 9.92·27-s + 2.82·28-s − 4.08·29-s − 9.92·31-s + 11.8·32-s − 7.83·33-s + 10.7·34-s + ⋯ |
| L(s) = 1 | + 1.85·2-s − 1.75·3-s + 2.45·4-s − 3.26·6-s + 0.217·7-s + 2.71·8-s + 2.08·9-s + 0.776·11-s − 4.31·12-s + 0.129·13-s + 0.403·14-s + 2.58·16-s + 0.991·17-s + 3.88·18-s + 0.229·19-s − 0.381·21-s + 1.44·22-s − 0.315·23-s − 4.76·24-s + 0.241·26-s − 1.90·27-s + 0.534·28-s − 0.758·29-s − 1.78·31-s + 2.09·32-s − 1.36·33-s + 1.84·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.714377895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.714377895\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.63T + 2T^{2} \) |
| 3 | \( 1 + 3.04T + 3T^{2} \) |
| 7 | \( 1 - 0.574T + 7T^{2} \) |
| 11 | \( 1 - 2.57T + 11T^{2} \) |
| 13 | \( 1 - 0.468T + 13T^{2} \) |
| 17 | \( 1 - 4.08T + 17T^{2} \) |
| 23 | \( 1 + 1.51T + 23T^{2} \) |
| 29 | \( 1 + 4.08T + 29T^{2} \) |
| 31 | \( 1 + 9.92T + 31T^{2} \) |
| 37 | \( 1 - 8.30T + 37T^{2} \) |
| 41 | \( 1 + 1.83T + 41T^{2} \) |
| 43 | \( 1 - 0.574T + 43T^{2} \) |
| 47 | \( 1 + 7.09T + 47T^{2} \) |
| 53 | \( 1 + 4.30T + 53T^{2} \) |
| 59 | \( 1 + 2.68T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 + 7.40T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + 6.68T + 79T^{2} \) |
| 83 | \( 1 - 6.66T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
| 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
−11.43895686553886567026166484210, −10.75834996822025074953302869866, −9.688766950972113384410357115116, −7.62731377401260901881907374390, −6.76894716193023402380030785973, −5.94629681735106961175506607795, −5.39030718820709127941209345339, −4.48014821404409193064830801712, −3.54161578949348247210801353318, −1.57834413627581354642474715405,
1.57834413627581354642474715405, 3.54161578949348247210801353318, 4.48014821404409193064830801712, 5.39030718820709127941209345339, 5.94629681735106961175506607795, 6.76894716193023402380030785973, 7.62731377401260901881907374390, 9.688766950972113384410357115116, 10.75834996822025074953302869866, 11.43895686553886567026166484210
