| L(s) = 1 | − 2.45·2-s + 3-s + 4.00·4-s + 2.55·5-s − 2.45·6-s − 7-s − 4.91·8-s + 9-s − 6.25·10-s − 4.54·11-s + 4.00·12-s − 1.98·13-s + 2.45·14-s + 2.55·15-s + 4.03·16-s − 0.0223·17-s − 2.45·18-s + 7.33·19-s + 10.2·20-s − 21-s + 11.1·22-s − 6.17·23-s − 4.91·24-s + 1.51·25-s + 4.86·26-s + 27-s − 4.00·28-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 0.577·3-s + 2.00·4-s + 1.14·5-s − 1.00·6-s − 0.377·7-s − 1.73·8-s + 0.333·9-s − 1.97·10-s − 1.37·11-s + 1.15·12-s − 0.550·13-s + 0.654·14-s + 0.658·15-s + 1.00·16-s − 0.00543·17-s − 0.577·18-s + 1.68·19-s + 2.28·20-s − 0.218·21-s + 2.37·22-s − 1.28·23-s − 1.00·24-s + 0.302·25-s + 0.953·26-s + 0.192·27-s − 0.756·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 + 2.45T + 2T^{2} \) |
| 5 | \( 1 - 2.55T + 5T^{2} \) |
| 11 | \( 1 + 4.54T + 11T^{2} \) |
| 13 | \( 1 + 1.98T + 13T^{2} \) |
| 17 | \( 1 + 0.0223T + 17T^{2} \) |
| 19 | \( 1 - 7.33T + 19T^{2} \) |
| 23 | \( 1 + 6.17T + 23T^{2} \) |
| 29 | \( 1 + 3.26T + 29T^{2} \) |
| 31 | \( 1 - 2.76T + 31T^{2} \) |
| 37 | \( 1 + 0.724T + 37T^{2} \) |
| 41 | \( 1 - 5.40T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 + 1.89T + 47T^{2} \) |
| 53 | \( 1 - 9.71T + 53T^{2} \) |
| 59 | \( 1 + 6.81T + 59T^{2} \) |
| 61 | \( 1 + 3.07T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 1.31T + 71T^{2} \) |
| 73 | \( 1 + 9.81T + 73T^{2} \) |
| 79 | \( 1 - 9.83T + 79T^{2} \) |
| 83 | \( 1 + 3.52T + 83T^{2} \) |
| 89 | \( 1 + 3.23T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−7.57530162294004626426320980388, −7.25986187131420692126065371711, −6.26442949115543334020386717716, −5.66627249520885007101924318427, −4.86514343581268124242589176685, −3.46722982666218787555924340598, −2.56441950892601911423753308219, −2.17843680713026229884972097619, −1.22075867715286844416547298800, 0,
1.22075867715286844416547298800, 2.17843680713026229884972097619, 2.56441950892601911423753308219, 3.46722982666218787555924340598, 4.86514343581268124242589176685, 5.66627249520885007101924318427, 6.26442949115543334020386717716, 7.25986187131420692126065371711, 7.57530162294004626426320980388
