| L(s) = 1 | − 0.207·2-s + 3-s − 1.95·4-s − 0.942·5-s − 0.207·6-s + 0.527·7-s + 0.820·8-s + 9-s + 0.195·10-s − 1.95·12-s − 13-s − 0.109·14-s − 0.942·15-s + 3.74·16-s + 3.52·17-s − 0.207·18-s + 6.90·19-s + 1.84·20-s + 0.527·21-s − 0.338·23-s + 0.820·24-s − 4.11·25-s + 0.207·26-s + 27-s − 1.03·28-s − 10.1·29-s + 0.195·30-s + ⋯ |
| L(s) = 1 | − 0.146·2-s + 0.577·3-s − 0.978·4-s − 0.421·5-s − 0.0846·6-s + 0.199·7-s + 0.290·8-s + 0.333·9-s + 0.0617·10-s − 0.564·12-s − 0.277·13-s − 0.0292·14-s − 0.243·15-s + 0.935·16-s + 0.855·17-s − 0.0488·18-s + 1.58·19-s + 0.412·20-s + 0.115·21-s − 0.0706·23-s + 0.167·24-s − 0.822·25-s + 0.0406·26-s + 0.192·27-s − 0.195·28-s − 1.87·29-s + 0.0356·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.578608053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.578608053\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 0.207T + 2T^{2} \) |
| 5 | \( 1 + 0.942T + 5T^{2} \) |
| 7 | \( 1 - 0.527T + 7T^{2} \) |
| 17 | \( 1 - 3.52T + 17T^{2} \) |
| 19 | \( 1 - 6.90T + 19T^{2} \) |
| 23 | \( 1 + 0.338T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 + 0.882T + 31T^{2} \) |
| 37 | \( 1 - 7.78T + 37T^{2} \) |
| 41 | \( 1 + 0.0708T + 41T^{2} \) |
| 43 | \( 1 - 5.35T + 43T^{2} \) |
| 47 | \( 1 - 0.682T + 47T^{2} \) |
| 53 | \( 1 + 4.80T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 2.76T + 71T^{2} \) |
| 73 | \( 1 - 1.03T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 4.59T + 97T^{2} \) |
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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
−8.186748832343061274259110518953, −7.66799750649722138924990521963, −7.31444802255288781755526567779, −5.90450167958801866654254878327, −5.32548921729223332314157595230, −4.47375917841844095663268261437, −3.71543744958346064056810634479, −3.12987679570851064324613022130, −1.81946800122648601322388017471, −0.71523257515034794867668267328,
0.71523257515034794867668267328, 1.81946800122648601322388017471, 3.12987679570851064324613022130, 3.71543744958346064056810634479, 4.47375917841844095663268261437, 5.32548921729223332314157595230, 5.90450167958801866654254878327, 7.31444802255288781755526567779, 7.66799750649722138924990521963, 8.186748832343061274259110518953
