L(s) = 1 | + 0.879·2-s − 3.22·3-s − 1.22·4-s + 5-s − 2.83·6-s − 3.71·7-s − 2.83·8-s + 7.41·9-s + 0.879·10-s + 1.18·11-s + 3.95·12-s + 1.57·13-s − 3.26·14-s − 3.22·15-s − 0.0418·16-s + 3.29·17-s + 6.51·18-s + 3·19-s − 1.22·20-s + 11.9·21-s + 1.04·22-s − 1.81·23-s + 9.15·24-s + 25-s + 1.38·26-s − 14.2·27-s + 4.55·28-s + ⋯ |
L(s) = 1 | + 0.621·2-s − 1.86·3-s − 0.613·4-s + 0.447·5-s − 1.15·6-s − 1.40·7-s − 1.00·8-s + 2.47·9-s + 0.278·10-s + 0.357·11-s + 1.14·12-s + 0.436·13-s − 0.873·14-s − 0.833·15-s − 0.0104·16-s + 0.798·17-s + 1.53·18-s + 0.688·19-s − 0.274·20-s + 2.61·21-s + 0.222·22-s − 0.378·23-s + 1.86·24-s + 0.200·25-s + 0.271·26-s − 2.73·27-s + 0.861·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 - 0.879T + 2T^{2} \) |
| 3 | \( 1 + 3.22T + 3T^{2} \) |
| 7 | \( 1 + 3.71T + 7T^{2} \) |
| 11 | \( 1 - 1.18T + 11T^{2} \) |
| 13 | \( 1 - 1.57T + 13T^{2} \) |
| 17 | \( 1 - 3.29T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 1.81T + 23T^{2} \) |
| 29 | \( 1 + 5.98T + 29T^{2} \) |
| 31 | \( 1 - 4.71T + 31T^{2} \) |
| 37 | \( 1 - 9.59T + 37T^{2} \) |
| 41 | \( 1 + 3.42T + 41T^{2} \) |
| 43 | \( 1 + 0.958T + 43T^{2} \) |
| 47 | \( 1 + 6.33T + 47T^{2} \) |
| 53 | \( 1 + 6.53T + 53T^{2} \) |
| 59 | \( 1 - 7.74T + 59T^{2} \) |
| 61 | \( 1 + 9.00T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 1.32T + 71T^{2} \) |
| 73 | \( 1 + 3.24T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 2.32T + 89T^{2} \) |
| 97 | \( 1 - 4.84T + 97T^{2} \) |
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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
−9.140006006380526809175144610848, −7.73859330831888787102031717519, −6.66708985608238381168019148160, −6.10867052571747491454125227798, −5.70969493732050399106857620737, −4.88051336665503568704528803359, −4.00502870866672297410832719925, −3.14499219088141689179084039309, −1.18241474268860804309603696190, 0,
1.18241474268860804309603696190, 3.14499219088141689179084039309, 4.00502870866672297410832719925, 4.88051336665503568704528803359, 5.70969493732050399106857620737, 6.10867052571747491454125227798, 6.66708985608238381168019148160, 7.73859330831888787102031717519, 9.140006006380526809175144610848