| L(s) = 1 | + 1.99·2-s + 0.909·3-s + 1.98·4-s + 5-s + 1.81·6-s + 1.84·7-s − 0.0225·8-s − 2.17·9-s + 1.99·10-s − 0.655·11-s + 1.80·12-s − 6.04·13-s + 3.69·14-s + 0.909·15-s − 4.02·16-s − 17-s − 4.33·18-s − 5.53·19-s + 1.98·20-s + 1.68·21-s − 1.30·22-s − 5.67·23-s − 0.0204·24-s + 25-s − 12.0·26-s − 4.70·27-s + 3.67·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.525·3-s + 0.994·4-s + 0.447·5-s + 0.741·6-s + 0.699·7-s − 0.00796·8-s − 0.724·9-s + 0.631·10-s − 0.197·11-s + 0.522·12-s − 1.67·13-s + 0.987·14-s + 0.234·15-s − 1.00·16-s − 0.242·17-s − 1.02·18-s − 1.27·19-s + 0.444·20-s + 0.367·21-s − 0.279·22-s − 1.18·23-s − 0.00418·24-s + 0.200·25-s − 2.36·26-s − 0.905·27-s + 0.695·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 2 | \( 1 - 1.99T + 2T^{2} \) |
| 3 | \( 1 - 0.909T + 3T^{2} \) |
| 7 | \( 1 - 1.84T + 7T^{2} \) |
| 11 | \( 1 + 0.655T + 11T^{2} \) |
| 13 | \( 1 + 6.04T + 13T^{2} \) |
| 19 | \( 1 + 5.53T + 19T^{2} \) |
| 23 | \( 1 + 5.67T + 23T^{2} \) |
| 29 | \( 1 - 5.48T + 29T^{2} \) |
| 31 | \( 1 - 1.49T + 31T^{2} \) |
| 37 | \( 1 + 1.83T + 37T^{2} \) |
| 41 | \( 1 - 1.06T + 41T^{2} \) |
| 43 | \( 1 - 8.12T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 + 5.09T + 53T^{2} \) |
| 59 | \( 1 - 6.14T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 6.16T + 67T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 4.92T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 5.63T + 97T^{2} \) |
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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
−7.83176092670834077922619828348, −6.66680602083915427944440015344, −6.22920779636309290845395194475, −5.31567714806903873495891480387, −4.82850127250684072465276629539, −4.21577070893623638346917305187, −3.23152098523976447899586459459, −2.41196981179682733816971524411, −2.05122881693703782005423337146, 0,
2.05122881693703782005423337146, 2.41196981179682733816971524411, 3.23152098523976447899586459459, 4.21577070893623638346917305187, 4.82850127250684072465276629539, 5.31567714806903873495891480387, 6.22920779636309290845395194475, 6.66680602083915427944440015344, 7.83176092670834077922619828348
