L(s) = 1 | − 2-s − 3-s + 4-s − 2.94·5-s + 6-s − 3.14·7-s − 8-s + 9-s + 2.94·10-s + 0.137·11-s − 12-s + 13-s + 3.14·14-s + 2.94·15-s + 16-s − 2.74·17-s − 18-s − 4.14·19-s − 2.94·20-s + 3.14·21-s − 0.137·22-s + 7.19·23-s + 24-s + 3.65·25-s − 26-s − 27-s − 3.14·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.31·5-s + 0.408·6-s − 1.19·7-s − 0.353·8-s + 0.333·9-s + 0.930·10-s + 0.0413·11-s − 0.288·12-s + 0.277·13-s + 0.841·14-s + 0.759·15-s + 0.250·16-s − 0.666·17-s − 0.235·18-s − 0.951·19-s − 0.657·20-s + 0.687·21-s − 0.0292·22-s + 1.49·23-s + 0.204·24-s + 0.730·25-s − 0.196·26-s − 0.192·27-s − 0.595·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 2.94T + 5T^{2} \) |
| 7 | \( 1 + 3.14T + 7T^{2} \) |
| 11 | \( 1 - 0.137T + 11T^{2} \) |
| 17 | \( 1 + 2.74T + 17T^{2} \) |
| 19 | \( 1 + 4.14T + 19T^{2} \) |
| 23 | \( 1 - 7.19T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 - 4.60T + 31T^{2} \) |
| 37 | \( 1 + 0.829T + 37T^{2} \) |
| 41 | \( 1 - 4.33T + 41T^{2} \) |
| 43 | \( 1 + 9.96T + 43T^{2} \) |
| 47 | \( 1 - 6.77T + 47T^{2} \) |
| 53 | \( 1 - 5.15T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 8.47T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 + 4.30T + 71T^{2} \) |
| 73 | \( 1 - 1.09T + 73T^{2} \) |
| 79 | \( 1 + 2.76T + 79T^{2} \) |
| 83 | \( 1 - 8.49T + 83T^{2} \) |
| 89 | \( 1 - 3.36T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−7.37111788504496307592901713120, −6.89976081891060835415015005620, −6.36695021056266406044981324674, −5.54403105756586853682163993291, −4.56005903007007039348089732863, −3.80768723178610052054885029856, −3.20797307415258400429063837591, −2.14840037383694094793760962160, −0.78019607539215726766331184144, 0,
0.78019607539215726766331184144, 2.14840037383694094793760962160, 3.20797307415258400429063837591, 3.80768723178610052054885029856, 4.56005903007007039348089732863, 5.54403105756586853682163993291, 6.36695021056266406044981324674, 6.89976081891060835415015005620, 7.37111788504496307592901713120