| L(s) = 1 | + 2-s − 2.41·3-s + 4-s + 4.21·5-s − 2.41·6-s − 1.11·7-s + 8-s + 2.82·9-s + 4.21·10-s − 5.09·11-s − 2.41·12-s − 6.06·13-s − 1.11·14-s − 10.1·15-s + 16-s − 4.52·17-s + 2.82·18-s − 7.95·19-s + 4.21·20-s + 2.68·21-s − 5.09·22-s + 23-s − 2.41·24-s + 12.7·25-s − 6.06·26-s + 0.414·27-s − 1.11·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.39·3-s + 0.5·4-s + 1.88·5-s − 0.985·6-s − 0.420·7-s + 0.353·8-s + 0.942·9-s + 1.33·10-s − 1.53·11-s − 0.696·12-s − 1.68·13-s − 0.297·14-s − 2.62·15-s + 0.250·16-s − 1.09·17-s + 0.666·18-s − 1.82·19-s + 0.941·20-s + 0.586·21-s − 1.08·22-s + 0.208·23-s − 0.492·24-s + 2.54·25-s − 1.19·26-s + 0.0797·27-s − 0.210·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 - 4.21T + 5T^{2} \) |
| 7 | \( 1 + 1.11T + 7T^{2} \) |
| 11 | \( 1 + 5.09T + 11T^{2} \) |
| 13 | \( 1 + 6.06T + 13T^{2} \) |
| 17 | \( 1 + 4.52T + 17T^{2} \) |
| 19 | \( 1 + 7.95T + 19T^{2} \) |
| 31 | \( 1 + 1.50T + 31T^{2} \) |
| 37 | \( 1 - 2.16T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 + 3.61T + 43T^{2} \) |
| 47 | \( 1 + 8.55T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 8.65T + 59T^{2} \) |
| 61 | \( 1 - 6.80T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 5.31T + 71T^{2} \) |
| 73 | \( 1 + 8.23T + 73T^{2} \) |
| 79 | \( 1 - 6.44T + 79T^{2} \) |
| 83 | \( 1 + 5.28T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 9.67T + 97T^{2} \) |
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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
−9.641065419469554030419962738746, −8.434145724776571258483407221349, −6.92431228401404024499414905552, −6.59111148282513384770543980515, −5.68455468315646047436700756959, −5.18547149338347379988993788918, −4.58757921889251636294135376938, −2.62685257265831172501045137190, −2.09059551672174916897093206384, 0,
2.09059551672174916897093206384, 2.62685257265831172501045137190, 4.58757921889251636294135376938, 5.18547149338347379988993788918, 5.68455468315646047436700756959, 6.59111148282513384770543980515, 6.92431228401404024499414905552, 8.434145724776571258483407221349, 9.641065419469554030419962738746
