| L(s) = 1 | + 1.74·2-s + 3-s + 1.04·4-s + 5-s + 1.74·6-s − 3.12·7-s − 1.66·8-s + 9-s + 1.74·10-s + 2.19·11-s + 1.04·12-s − 13-s − 5.45·14-s + 15-s − 4.99·16-s − 4.53·17-s + 1.74·18-s + 1.65·19-s + 1.04·20-s − 3.12·21-s + 3.83·22-s − 0.531·23-s − 1.66·24-s + 25-s − 1.74·26-s + 27-s − 3.27·28-s + ⋯ |
| L(s) = 1 | + 1.23·2-s + 0.577·3-s + 0.523·4-s + 0.447·5-s + 0.712·6-s − 1.18·7-s − 0.588·8-s + 0.333·9-s + 0.551·10-s + 0.662·11-s + 0.302·12-s − 0.277·13-s − 1.45·14-s + 0.258·15-s − 1.24·16-s − 1.09·17-s + 0.411·18-s + 0.378·19-s + 0.234·20-s − 0.682·21-s + 0.817·22-s − 0.110·23-s − 0.339·24-s + 0.200·25-s − 0.342·26-s + 0.192·27-s − 0.618·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 1.74T + 2T^{2} \) |
| 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 - 2.19T + 11T^{2} \) |
| 17 | \( 1 + 4.53T + 17T^{2} \) |
| 19 | \( 1 - 1.65T + 19T^{2} \) |
| 23 | \( 1 + 0.531T + 23T^{2} \) |
| 29 | \( 1 + 2.11T + 29T^{2} \) |
| 37 | \( 1 + 7.76T + 37T^{2} \) |
| 41 | \( 1 + 0.346T + 41T^{2} \) |
| 43 | \( 1 + 0.751T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 - 1.10T + 59T^{2} \) |
| 61 | \( 1 + 4.38T + 61T^{2} \) |
| 67 | \( 1 + 5.34T + 67T^{2} \) |
| 71 | \( 1 + 7.82T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 8.60T + 79T^{2} \) |
| 83 | \( 1 + 8.27T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−7.39103117655922044656784866705, −6.83144165331973010424868534923, −6.18062612764810015715713984348, −5.61514968190713206678834905498, −4.64405343751532345762835401915, −4.04728195345881924041602708595, −3.26052861260940889243993866368, −2.72879249005293419365055704242, −1.72164198999494644244538349344, 0,
1.72164198999494644244538349344, 2.72879249005293419365055704242, 3.26052861260940889243993866368, 4.04728195345881924041602708595, 4.64405343751532345762835401915, 5.61514968190713206678834905498, 6.18062612764810015715713984348, 6.83144165331973010424868534923, 7.39103117655922044656784866705
