| L(s) = 1 | − 1.38·2-s + 2.82·3-s − 0.0791·4-s + 0.518·5-s − 3.91·6-s + 2.88·8-s + 4.98·9-s − 0.719·10-s − 1.62·11-s − 0.223·12-s + 1.46·15-s − 3.83·16-s − 1.94·17-s − 6.90·18-s + 2.49·19-s − 0.0410·20-s + 2.25·22-s − 9.14·23-s + 8.14·24-s − 4.73·25-s + 5.60·27-s − 5.22·29-s − 2.03·30-s + 5.79·31-s − 0.447·32-s − 4.58·33-s + 2.70·34-s + ⋯ |
| L(s) = 1 | − 0.980·2-s + 1.63·3-s − 0.0395·4-s + 0.232·5-s − 1.59·6-s + 1.01·8-s + 1.66·9-s − 0.227·10-s − 0.489·11-s − 0.0645·12-s + 0.378·15-s − 0.958·16-s − 0.472·17-s − 1.62·18-s + 0.571·19-s − 0.00918·20-s + 0.479·22-s − 1.90·23-s + 1.66·24-s − 0.946·25-s + 1.07·27-s − 0.971·29-s − 0.371·30-s + 1.04·31-s − 0.0790·32-s − 0.798·33-s + 0.463·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.38T + 2T^{2} \) |
| 3 | \( 1 - 2.82T + 3T^{2} \) |
| 5 | \( 1 - 0.518T + 5T^{2} \) |
| 11 | \( 1 + 1.62T + 11T^{2} \) |
| 17 | \( 1 + 1.94T + 17T^{2} \) |
| 19 | \( 1 - 2.49T + 19T^{2} \) |
| 23 | \( 1 + 9.14T + 23T^{2} \) |
| 29 | \( 1 + 5.22T + 29T^{2} \) |
| 31 | \( 1 - 5.79T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 4.20T + 41T^{2} \) |
| 43 | \( 1 + 0.997T + 43T^{2} \) |
| 47 | \( 1 + 4.51T + 47T^{2} \) |
| 53 | \( 1 + 8.89T + 53T^{2} \) |
| 59 | \( 1 - 6.20T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 8.37T + 67T^{2} \) |
| 71 | \( 1 - 5.19T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 0.982T + 79T^{2} \) |
| 83 | \( 1 + 8.91T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 - 4.42T + 97T^{2} \) |
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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
−7.84064317071720407246369909467, −7.24155833697129203187523572965, −6.28682769426025525096617009426, −5.35276241809911459389062757487, −4.30268661213362869590398978194, −3.89529424317487692721605175534, −2.83651920245197985286762686248, −2.12969152065884468852853984017, −1.44092533288954876886338514966, 0,
1.44092533288954876886338514966, 2.12969152065884468852853984017, 2.83651920245197985286762686248, 3.89529424317487692721605175534, 4.30268661213362869590398978194, 5.35276241809911459389062757487, 6.28682769426025525096617009426, 7.24155833697129203187523572965, 7.84064317071720407246369909467
