| L(s) = 1 | − 2.37·2-s + 3-s + 3.64·4-s + 1.54·5-s − 2.37·6-s + 1.30·7-s − 3.91·8-s + 9-s − 3.68·10-s + 0.998·11-s + 3.64·12-s − 13-s − 3.09·14-s + 1.54·15-s + 2.00·16-s − 6.18·17-s − 2.37·18-s − 3.83·19-s + 5.65·20-s + 1.30·21-s − 2.37·22-s + 3.79·23-s − 3.91·24-s − 2.60·25-s + 2.37·26-s + 27-s + 4.75·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 0.577·3-s + 1.82·4-s + 0.692·5-s − 0.970·6-s + 0.492·7-s − 1.38·8-s + 0.333·9-s − 1.16·10-s + 0.300·11-s + 1.05·12-s − 0.277·13-s − 0.828·14-s + 0.399·15-s + 0.502·16-s − 1.50·17-s − 0.560·18-s − 0.879·19-s + 1.26·20-s + 0.284·21-s − 0.505·22-s + 0.792·23-s − 0.799·24-s − 0.520·25-s + 0.466·26-s + 0.192·27-s + 0.899·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 2.37T + 2T^{2} \) |
| 5 | \( 1 - 1.54T + 5T^{2} \) |
| 7 | \( 1 - 1.30T + 7T^{2} \) |
| 11 | \( 1 - 0.998T + 11T^{2} \) |
| 17 | \( 1 + 6.18T + 17T^{2} \) |
| 19 | \( 1 + 3.83T + 19T^{2} \) |
| 23 | \( 1 - 3.79T + 23T^{2} \) |
| 29 | \( 1 - 2.72T + 29T^{2} \) |
| 31 | \( 1 + 3.88T + 31T^{2} \) |
| 37 | \( 1 + 4.36T + 37T^{2} \) |
| 41 | \( 1 - 1.99T + 41T^{2} \) |
| 43 | \( 1 + 4.74T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 1.58T + 53T^{2} \) |
| 59 | \( 1 + 9.55T + 59T^{2} \) |
| 61 | \( 1 - 0.815T + 61T^{2} \) |
| 67 | \( 1 + 8.73T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 3.23T + 73T^{2} \) |
| 79 | \( 1 + 8.61T + 79T^{2} \) |
| 83 | \( 1 - 7.32T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 7.89T + 97T^{2} \) |
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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
−8.293448897104103842671753317005, −7.62518265863380971097691178845, −6.73812806790590290403849349067, −6.40303556015147745332977457565, −5.10438700583221415686100460639, −4.24196694675607571250577674437, −2.92132518857813646943795520385, −2.01636507994009584976808163806, −1.54560477655518194496459803156, 0,
1.54560477655518194496459803156, 2.01636507994009584976808163806, 2.92132518857813646943795520385, 4.24196694675607571250577674437, 5.10438700583221415686100460639, 6.40303556015147745332977457565, 6.73812806790590290403849349067, 7.62518265863380971097691178845, 8.293448897104103842671753317005
