L(s) = 1 | − 2.46·2-s + 2.70·3-s + 4.06·4-s + 0.207·5-s − 6.67·6-s − 2.27·7-s − 5.09·8-s + 4.34·9-s − 0.510·10-s + 0.737·11-s + 11.0·12-s − 1.97·13-s + 5.61·14-s + 0.561·15-s + 4.42·16-s − 7.55·17-s − 10.6·18-s + 5.52·19-s + 0.844·20-s − 6.17·21-s − 1.81·22-s + 2.66·23-s − 13.8·24-s − 4.95·25-s + 4.86·26-s + 3.63·27-s − 9.27·28-s + ⋯ |
L(s) = 1 | − 1.74·2-s + 1.56·3-s + 2.03·4-s + 0.0927·5-s − 2.72·6-s − 0.861·7-s − 1.80·8-s + 1.44·9-s − 0.161·10-s + 0.222·11-s + 3.18·12-s − 0.547·13-s + 1.50·14-s + 0.145·15-s + 1.10·16-s − 1.83·17-s − 2.52·18-s + 1.26·19-s + 0.188·20-s − 1.34·21-s − 0.387·22-s + 0.554·23-s − 2.81·24-s − 0.991·25-s + 0.954·26-s + 0.698·27-s − 1.75·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 6047 | \( 1+O(T) \) |
good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 3 | \( 1 - 2.70T + 3T^{2} \) |
| 5 | \( 1 - 0.207T + 5T^{2} \) |
| 7 | \( 1 + 2.27T + 7T^{2} \) |
| 11 | \( 1 - 0.737T + 11T^{2} \) |
| 13 | \( 1 + 1.97T + 13T^{2} \) |
| 17 | \( 1 + 7.55T + 17T^{2} \) |
| 19 | \( 1 - 5.52T + 19T^{2} \) |
| 23 | \( 1 - 2.66T + 23T^{2} \) |
| 29 | \( 1 - 7.95T + 29T^{2} \) |
| 31 | \( 1 + 7.15T + 31T^{2} \) |
| 37 | \( 1 - 6.07T + 37T^{2} \) |
| 41 | \( 1 + 2.92T + 41T^{2} \) |
| 43 | \( 1 - 4.05T + 43T^{2} \) |
| 47 | \( 1 + 4.70T + 47T^{2} \) |
| 53 | \( 1 + 1.50T + 53T^{2} \) |
| 59 | \( 1 - 5.97T + 59T^{2} \) |
| 61 | \( 1 - 2.80T + 61T^{2} \) |
| 67 | \( 1 - 2.28T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 3.04T + 73T^{2} \) |
| 79 | \( 1 + 5.42T + 79T^{2} \) |
| 83 | \( 1 - 4.78T + 83T^{2} \) |
| 89 | \( 1 + 8.65T + 89T^{2} \) |
| 97 | \( 1 + 6.74T + 97T^{2} \) |
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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
−7.979757727811718338740808904468, −7.17855417318648901768468056919, −6.90432421434425006915810177429, −5.99913802941272976019262197469, −4.64098362483410810454371198334, −3.62837856200453527738498207475, −2.77399864623331293314182960829, −2.30775325836067772757506980480, −1.34288977615824213037942568270, 0,
1.34288977615824213037942568270, 2.30775325836067772757506980480, 2.77399864623331293314182960829, 3.62837856200453527738498207475, 4.64098362483410810454371198334, 5.99913802941272976019262197469, 6.90432421434425006915810177429, 7.17855417318648901768468056919, 7.979757727811718338740808904468