| L(s) = 1 | + 2-s + 1.80·3-s + 4-s + 0.400·5-s + 1.80·6-s − 2.14·7-s + 8-s + 0.243·9-s + 0.400·10-s − 4.21·11-s + 1.80·12-s + 4.87·13-s − 2.14·14-s + 0.721·15-s + 16-s − 3.84·17-s + 0.243·18-s − 19-s + 0.400·20-s − 3.85·21-s − 4.21·22-s − 2.95·23-s + 1.80·24-s − 4.83·25-s + 4.87·26-s − 4.96·27-s − 2.14·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.03·3-s + 0.5·4-s + 0.179·5-s + 0.735·6-s − 0.809·7-s + 0.353·8-s + 0.0810·9-s + 0.126·10-s − 1.26·11-s + 0.519·12-s + 1.35·13-s − 0.572·14-s + 0.186·15-s + 0.250·16-s − 0.933·17-s + 0.0573·18-s − 0.229·19-s + 0.0896·20-s − 0.841·21-s − 0.897·22-s − 0.616·23-s + 0.367·24-s − 0.967·25-s + 0.956·26-s − 0.955·27-s − 0.404·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 1.80T + 3T^{2} \) |
| 5 | \( 1 - 0.400T + 5T^{2} \) |
| 7 | \( 1 + 2.14T + 7T^{2} \) |
| 11 | \( 1 + 4.21T + 11T^{2} \) |
| 13 | \( 1 - 4.87T + 13T^{2} \) |
| 17 | \( 1 + 3.84T + 17T^{2} \) |
| 23 | \( 1 + 2.95T + 23T^{2} \) |
| 29 | \( 1 - 4.33T + 29T^{2} \) |
| 31 | \( 1 + 0.370T + 31T^{2} \) |
| 37 | \( 1 + 4.56T + 37T^{2} \) |
| 41 | \( 1 + 2.73T + 41T^{2} \) |
| 43 | \( 1 - 9.17T + 43T^{2} \) |
| 47 | \( 1 - 4.26T + 47T^{2} \) |
| 53 | \( 1 + 3.99T + 53T^{2} \) |
| 59 | \( 1 + 3.00T + 59T^{2} \) |
| 61 | \( 1 + 5.26T + 61T^{2} \) |
| 67 | \( 1 + 1.14T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 + 6.07T + 73T^{2} \) |
| 79 | \( 1 + 1.43T + 79T^{2} \) |
| 83 | \( 1 + 1.24T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−7.63119456722853359965755267603, −6.64854767757881662419921966707, −6.05709235562200742061907448233, −5.52088983729448056416758639340, −4.44880870751428557062165508606, −3.79781683795389578418896666923, −3.06441119420441688156971509654, −2.53162295088827183607819271328, −1.69250173005833210440201477172, 0,
1.69250173005833210440201477172, 2.53162295088827183607819271328, 3.06441119420441688156971509654, 3.79781683795389578418896666923, 4.44880870751428557062165508606, 5.52088983729448056416758639340, 6.05709235562200742061907448233, 6.64854767757881662419921966707, 7.63119456722853359965755267603
