| L(s) = 1 | − 2-s − 0.221·3-s + 4-s − 0.411·5-s + 0.221·6-s − 7-s − 8-s − 2.95·9-s + 0.411·10-s + 2.69·11-s − 0.221·12-s + 1.03·13-s + 14-s + 0.0911·15-s + 16-s + 3.23·17-s + 2.95·18-s + 4.82·19-s − 0.411·20-s + 0.221·21-s − 2.69·22-s − 3.50·23-s + 0.221·24-s − 4.83·25-s − 1.03·26-s + 1.32·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.128·3-s + 0.5·4-s − 0.183·5-s + 0.0905·6-s − 0.377·7-s − 0.353·8-s − 0.983·9-s + 0.129·10-s + 0.813·11-s − 0.0640·12-s + 0.286·13-s + 0.267·14-s + 0.0235·15-s + 0.250·16-s + 0.783·17-s + 0.695·18-s + 1.10·19-s − 0.0919·20-s + 0.0484·21-s − 0.575·22-s − 0.731·23-s + 0.0452·24-s − 0.966·25-s − 0.202·26-s + 0.254·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 0.221T + 3T^{2} \) |
| 5 | \( 1 + 0.411T + 5T^{2} \) |
| 11 | \( 1 - 2.69T + 11T^{2} \) |
| 13 | \( 1 - 1.03T + 13T^{2} \) |
| 17 | \( 1 - 3.23T + 17T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 + 3.50T + 23T^{2} \) |
| 29 | \( 1 - 6.47T + 29T^{2} \) |
| 31 | \( 1 + 9.99T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 - 5.63T + 41T^{2} \) |
| 43 | \( 1 + 2.15T + 43T^{2} \) |
| 47 | \( 1 + 1.43T + 47T^{2} \) |
| 53 | \( 1 - 8.04T + 53T^{2} \) |
| 59 | \( 1 + 4.90T + 59T^{2} \) |
| 61 | \( 1 + 9.82T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + 9.72T + 71T^{2} \) |
| 73 | \( 1 + 3.14T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 + 4.93T + 83T^{2} \) |
| 89 | \( 1 - 2.08T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−7.69738018189720416400717353337, −7.20669968391232452173128263265, −6.25361111118739690042245999290, −5.80631609098697153199709692568, −5.00864123176663170832993894529, −3.67301644354707237543884183758, −3.34593506267664622421272600788, −2.18687995578629527191872577543, −1.15841402988878183331139728430, 0,
1.15841402988878183331139728430, 2.18687995578629527191872577543, 3.34593506267664622421272600788, 3.67301644354707237543884183758, 5.00864123176663170832993894529, 5.80631609098697153199709692568, 6.25361111118739690042245999290, 7.20669968391232452173128263265, 7.69738018189720416400717353337
