| L(s) = 1 | + 1.15·2-s + 1.72·3-s − 0.658·4-s + 1.50·5-s + 1.99·6-s − 1.44·7-s − 3.07·8-s − 0.0220·9-s + 1.74·10-s + 3.52·11-s − 1.13·12-s − 1.67·14-s + 2.59·15-s − 2.25·16-s − 7.47·17-s − 0.0255·18-s + 1.72·19-s − 0.991·20-s − 2.49·21-s + 4.08·22-s − 5.29·23-s − 5.31·24-s − 2.73·25-s − 5.21·27-s + 0.950·28-s + 1.27·29-s + 3.01·30-s + ⋯ |
| L(s) = 1 | + 0.819·2-s + 0.996·3-s − 0.329·4-s + 0.673·5-s + 0.816·6-s − 0.545·7-s − 1.08·8-s − 0.00735·9-s + 0.551·10-s + 1.06·11-s − 0.327·12-s − 0.447·14-s + 0.671·15-s − 0.562·16-s − 1.81·17-s − 0.00602·18-s + 0.395·19-s − 0.221·20-s − 0.543·21-s + 0.870·22-s − 1.10·23-s − 1.08·24-s − 0.546·25-s − 1.00·27-s + 0.179·28-s + 0.237·29-s + 0.549·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 1.15T + 2T^{2} \) |
| 3 | \( 1 - 1.72T + 3T^{2} \) |
| 5 | \( 1 - 1.50T + 5T^{2} \) |
| 7 | \( 1 + 1.44T + 7T^{2} \) |
| 11 | \( 1 - 3.52T + 11T^{2} \) |
| 17 | \( 1 + 7.47T + 17T^{2} \) |
| 19 | \( 1 - 1.72T + 19T^{2} \) |
| 23 | \( 1 + 5.29T + 23T^{2} \) |
| 29 | \( 1 - 1.27T + 29T^{2} \) |
| 37 | \( 1 + 2.04T + 37T^{2} \) |
| 41 | \( 1 + 2.89T + 41T^{2} \) |
| 43 | \( 1 - 7.62T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 9.49T + 61T^{2} \) |
| 67 | \( 1 - 6.29T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 4.38T + 73T^{2} \) |
| 79 | \( 1 - 5.57T + 79T^{2} \) |
| 83 | \( 1 + 6.64T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 8.96T + 97T^{2} \) |
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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
−7.996296960497969345502934721711, −6.92516403801949084400048582037, −6.22053736593625287779339634182, −5.78840837706648811782099386107, −4.70348501121305233802658272932, −4.02507590063634585078588324180, −3.40224467738708735445291690128, −2.56923004963298503325693733436, −1.77759891930102724066042348704, 0,
1.77759891930102724066042348704, 2.56923004963298503325693733436, 3.40224467738708735445291690128, 4.02507590063634585078588324180, 4.70348501121305233802658272932, 5.78840837706648811782099386107, 6.22053736593625287779339634182, 6.92516403801949084400048582037, 7.996296960497969345502934721711
