| L(s) = 1 | + 0.336·2-s − 2.12·3-s − 1.88·4-s − 5-s − 0.712·6-s − 1.25·7-s − 1.30·8-s + 1.49·9-s − 0.336·10-s + 4.00·12-s − 13-s − 0.422·14-s + 2.12·15-s + 3.33·16-s + 1.59·17-s + 0.503·18-s + 6.34·19-s + 1.88·20-s + 2.66·21-s + 2.01·23-s + 2.77·24-s + 25-s − 0.336·26-s + 3.18·27-s + 2.37·28-s − 3.54·29-s + 0.712·30-s + ⋯ |
| L(s) = 1 | + 0.237·2-s − 1.22·3-s − 0.943·4-s − 0.447·5-s − 0.291·6-s − 0.475·7-s − 0.461·8-s + 0.499·9-s − 0.106·10-s + 1.15·12-s − 0.277·13-s − 0.112·14-s + 0.547·15-s + 0.833·16-s + 0.387·17-s + 0.118·18-s + 1.45·19-s + 0.421·20-s + 0.581·21-s + 0.420·23-s + 0.565·24-s + 0.200·25-s − 0.0659·26-s + 0.613·27-s + 0.448·28-s − 0.657·29-s + 0.130·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4351699960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4351699960\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 0.336T + 2T^{2} \) |
| 3 | \( 1 + 2.12T + 3T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 17 | \( 1 - 1.59T + 17T^{2} \) |
| 19 | \( 1 - 6.34T + 19T^{2} \) |
| 23 | \( 1 - 2.01T + 23T^{2} \) |
| 29 | \( 1 + 3.54T + 29T^{2} \) |
| 31 | \( 1 + 4.93T + 31T^{2} \) |
| 37 | \( 1 + 2.92T + 37T^{2} \) |
| 41 | \( 1 - 0.127T + 41T^{2} \) |
| 43 | \( 1 + 6.69T + 43T^{2} \) |
| 47 | \( 1 + 6.75T + 47T^{2} \) |
| 53 | \( 1 - 4.99T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 2.78T + 61T^{2} \) |
| 67 | \( 1 + 0.638T + 67T^{2} \) |
| 71 | \( 1 - 6.26T + 71T^{2} \) |
| 73 | \( 1 + 5.31T + 73T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 + 9.60T + 83T^{2} \) |
| 89 | \( 1 - 7.67T + 89T^{2} \) |
| 97 | \( 1 - 9.78T + 97T^{2} \) |
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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
−7.70799247407580357723582275048, −7.11335280660945800004738743258, −6.28966069209227207116956619707, −5.57111131863443653754335838507, −5.14171474558441244942559680302, −4.53134026415660439138735370229, −3.52765740790389236078217354734, −3.08331055617778576778748995689, −1.40330967008780789283774305152, −0.36429946897431506640744254740,
0.36429946897431506640744254740, 1.40330967008780789283774305152, 3.08331055617778576778748995689, 3.52765740790389236078217354734, 4.53134026415660439138735370229, 5.14171474558441244942559680302, 5.57111131863443653754335838507, 6.28966069209227207116956619707, 7.11335280660945800004738743258, 7.70799247407580357723582275048
