L(s) = 1 | − 1.35·2-s − 0.160·4-s − 1.12·5-s − 1.90·7-s + 2.93·8-s + 1.53·10-s − 2.12·11-s + 5.22·13-s + 2.57·14-s − 3.65·16-s + 2.05·17-s + 8.06·19-s + 0.181·20-s + 2.87·22-s + 0.751·23-s − 3.72·25-s − 7.09·26-s + 0.306·28-s + 8.06·29-s − 6.52·31-s − 0.908·32-s − 2.79·34-s + 2.14·35-s + 7.00·37-s − 10.9·38-s − 3.30·40-s − 6.68·41-s + ⋯ |
L(s) = 1 | − 0.958·2-s − 0.0804·4-s − 0.505·5-s − 0.718·7-s + 1.03·8-s + 0.484·10-s − 0.639·11-s + 1.45·13-s + 0.689·14-s − 0.913·16-s + 0.499·17-s + 1.85·19-s + 0.0406·20-s + 0.613·22-s + 0.156·23-s − 0.744·25-s − 1.39·26-s + 0.0578·28-s + 1.49·29-s − 1.17·31-s − 0.160·32-s − 0.478·34-s + 0.362·35-s + 1.15·37-s − 1.77·38-s − 0.523·40-s − 1.04·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8585785739\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8585785739\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 223 | \( 1 - T \) |
good | 2 | \( 1 + 1.35T + 2T^{2} \) |
| 5 | \( 1 + 1.12T + 5T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 11 | \( 1 + 2.12T + 11T^{2} \) |
| 13 | \( 1 - 5.22T + 13T^{2} \) |
| 17 | \( 1 - 2.05T + 17T^{2} \) |
| 19 | \( 1 - 8.06T + 19T^{2} \) |
| 23 | \( 1 - 0.751T + 23T^{2} \) |
| 29 | \( 1 - 8.06T + 29T^{2} \) |
| 31 | \( 1 + 6.52T + 31T^{2} \) |
| 37 | \( 1 - 7.00T + 37T^{2} \) |
| 41 | \( 1 + 6.68T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 6.43T + 61T^{2} \) |
| 67 | \( 1 + 3.01T + 67T^{2} \) |
| 71 | \( 1 - 6.13T + 71T^{2} \) |
| 73 | \( 1 - 7.57T + 73T^{2} \) |
| 79 | \( 1 + 6.18T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 - 2.52T + 89T^{2} \) |
| 97 | \( 1 - 1.13T + 97T^{2} \) |
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
−7.938935014882165034954869428157, −7.73636026898910460754632767225, −6.90471809810385157149514365471, −5.97268581017164746575693697354, −5.32022787065389027209325845144, −4.32399564551688519802027174844, −3.57257524000791239851176514967, −2.84754794290726707454445434763, −1.41709854100342799093270275427, −0.62476725839363017907972813539,
0.62476725839363017907972813539, 1.41709854100342799093270275427, 2.84754794290726707454445434763, 3.57257524000791239851176514967, 4.32399564551688519802027174844, 5.32022787065389027209325845144, 5.97268581017164746575693697354, 6.90471809810385157149514365471, 7.73636026898910460754632767225, 7.938935014882165034954869428157