L(s) = 1 | + 2-s + 3-s + 4-s − 0.877·5-s + 6-s − 4.30·7-s + 8-s + 9-s − 0.877·10-s + 2.30·11-s + 12-s + 5.05·13-s − 4.30·14-s − 0.877·15-s + 16-s + 17-s + 18-s + 6.89·19-s − 0.877·20-s − 4.30·21-s + 2.30·22-s − 3.74·23-s + 24-s − 4.22·25-s + 5.05·26-s + 27-s − 4.30·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.392·5-s + 0.408·6-s − 1.62·7-s + 0.353·8-s + 0.333·9-s − 0.277·10-s + 0.694·11-s + 0.288·12-s + 1.40·13-s − 1.15·14-s − 0.226·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 1.58·19-s − 0.196·20-s − 0.939·21-s + 0.491·22-s − 0.779·23-s + 0.204·24-s − 0.845·25-s + 0.990·26-s + 0.192·27-s − 0.813·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.439253479\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.439253479\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 + 0.877T + 5T^{2} \) |
| 7 | \( 1 + 4.30T + 7T^{2} \) |
| 11 | \( 1 - 2.30T + 11T^{2} \) |
| 13 | \( 1 - 5.05T + 13T^{2} \) |
| 19 | \( 1 - 6.89T + 19T^{2} \) |
| 23 | \( 1 + 3.74T + 23T^{2} \) |
| 29 | \( 1 + 2.11T + 29T^{2} \) |
| 31 | \( 1 + 3.57T + 31T^{2} \) |
| 37 | \( 1 + 5.04T + 37T^{2} \) |
| 41 | \( 1 - 2.42T + 41T^{2} \) |
| 43 | \( 1 - 1.15T + 43T^{2} \) |
| 47 | \( 1 - 4.76T + 47T^{2} \) |
| 53 | \( 1 - 9.00T + 53T^{2} \) |
| 61 | \( 1 + 0.436T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 3.56T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 2.70T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 0.796T + 97T^{2} \) |
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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
−7.898977022339792365913008659253, −7.31623143587137642204203757041, −6.56060646455685595179055012607, −6.00334210096119185563036081745, −5.32222819182980253873866587105, −3.92729115552022449541670656471, −3.71028396167220395068881925634, −3.17772728069016157264480571825, −2.06499396445502475338948747714, −0.861488612422071777307320967095,
0.861488612422071777307320967095, 2.06499396445502475338948747714, 3.17772728069016157264480571825, 3.71028396167220395068881925634, 3.92729115552022449541670656471, 5.32222819182980253873866587105, 6.00334210096119185563036081745, 6.56060646455685595179055012607, 7.31623143587137642204203757041, 7.898977022339792365913008659253