L(s) = 1 | + 1.81·2-s + 1.25·3-s + 1.30·4-s + 0.160·5-s + 2.28·6-s − 4.31·7-s − 1.25·8-s − 1.42·9-s + 0.291·10-s − 0.148·11-s + 1.64·12-s + 1.68·13-s − 7.85·14-s + 0.201·15-s − 4.90·16-s − 5.73·17-s − 2.58·18-s − 4.22·19-s + 0.210·20-s − 5.42·21-s − 0.269·22-s − 3.75·23-s − 1.57·24-s − 4.97·25-s + 3.06·26-s − 5.55·27-s − 5.65·28-s + ⋯ |
L(s) = 1 | + 1.28·2-s + 0.725·3-s + 0.654·4-s + 0.0717·5-s + 0.932·6-s − 1.63·7-s − 0.443·8-s − 0.474·9-s + 0.0922·10-s − 0.0447·11-s + 0.474·12-s + 0.467·13-s − 2.09·14-s + 0.0520·15-s − 1.22·16-s − 1.39·17-s − 0.610·18-s − 0.970·19-s + 0.0469·20-s − 1.18·21-s − 0.0575·22-s − 0.782·23-s − 0.321·24-s − 0.994·25-s + 0.601·26-s − 1.06·27-s − 1.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
good | 2 | \( 1 - 1.81T + 2T^{2} \) |
| 3 | \( 1 - 1.25T + 3T^{2} \) |
| 5 | \( 1 - 0.160T + 5T^{2} \) |
| 7 | \( 1 + 4.31T + 7T^{2} \) |
| 11 | \( 1 + 0.148T + 11T^{2} \) |
| 13 | \( 1 - 1.68T + 13T^{2} \) |
| 17 | \( 1 + 5.73T + 17T^{2} \) |
| 19 | \( 1 + 4.22T + 19T^{2} \) |
| 23 | \( 1 + 3.75T + 23T^{2} \) |
| 29 | \( 1 - 8.83T + 29T^{2} \) |
| 31 | \( 1 - 8.64T + 31T^{2} \) |
| 37 | \( 1 - 7.19T + 37T^{2} \) |
| 41 | \( 1 + 3.67T + 41T^{2} \) |
| 47 | \( 1 + 2.13T + 47T^{2} \) |
| 53 | \( 1 - 1.98T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 5.65T + 61T^{2} \) |
| 67 | \( 1 + 1.42T + 67T^{2} \) |
| 71 | \( 1 + 7.56T + 71T^{2} \) |
| 73 | \( 1 + 4.99T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 + 6.11T + 83T^{2} \) |
| 89 | \( 1 + 8.33T + 89T^{2} \) |
| 97 | \( 1 - 2.10T + 97T^{2} \) |
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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
−8.725409259368028179725072040623, −8.308171131400229269530196447175, −6.81732410200332217347569382459, −6.31237913800183101362158747325, −5.76637651124815879496041184640, −4.40839302688389479760610653307, −3.89668568113519097522264529689, −2.88373116920691727135570331062, −2.44361333361853109199474968952, 0,
2.44361333361853109199474968952, 2.88373116920691727135570331062, 3.89668568113519097522264529689, 4.40839302688389479760610653307, 5.76637651124815879496041184640, 6.31237913800183101362158747325, 6.81732410200332217347569382459, 8.308171131400229269530196447175, 8.725409259368028179725072040623