| L(s) = 1 | + 2-s − 0.381·3-s + 4-s − 4.23·5-s − 0.381·6-s − 7-s + 8-s − 2.85·9-s − 4.23·10-s + 3·11-s − 0.381·12-s + 3.85·13-s − 14-s + 1.61·15-s + 16-s + 1.85·17-s − 2.85·18-s − 4.23·20-s + 0.381·21-s + 3·22-s − 6.70·23-s − 0.381·24-s + 12.9·25-s + 3.85·26-s + 2.23·27-s − 28-s + 8.23·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.220·3-s + 0.5·4-s − 1.89·5-s − 0.155·6-s − 0.377·7-s + 0.353·8-s − 0.951·9-s − 1.33·10-s + 0.904·11-s − 0.110·12-s + 1.06·13-s − 0.267·14-s + 0.417·15-s + 0.250·16-s + 0.449·17-s − 0.672·18-s − 0.947·20-s + 0.0833·21-s + 0.639·22-s − 1.39·23-s − 0.0779·24-s + 2.58·25-s + 0.755·26-s + 0.430·27-s − 0.188·28-s + 1.52·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.381T + 3T^{2} \) |
| 5 | \( 1 + 4.23T + 5T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 3.85T + 13T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 23 | \( 1 + 6.70T + 23T^{2} \) |
| 29 | \( 1 - 8.23T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 - 4.09T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 6.23T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 6.70T + 61T^{2} \) |
| 67 | \( 1 - 4.56T + 67T^{2} \) |
| 71 | \( 1 + 3.52T + 71T^{2} \) |
| 73 | \( 1 + 9.09T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 6.47T + 83T^{2} \) |
| 89 | \( 1 - 1.38T + 89T^{2} \) |
| 97 | \( 1 - 5.29T + 97T^{2} \) |
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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
−7.935231853611193153903462142729, −7.04309238014470053871464462233, −6.36959955823973951078716651942, −5.78340018747528314465134444622, −4.71849670704738387897739752896, −4.02668717735028189981736651053, −3.50737109255495899081237430264, −2.84421019872987466891801361875, −1.23444376252803623352156962715, 0,
1.23444376252803623352156962715, 2.84421019872987466891801361875, 3.50737109255495899081237430264, 4.02668717735028189981736651053, 4.71849670704738387897739752896, 5.78340018747528314465134444622, 6.36959955823973951078716651942, 7.04309238014470053871464462233, 7.935231853611193153903462142729
