| L(s) = 1 | + 1.66·3-s + 5-s − 7-s − 0.236·9-s + 2.60·11-s − 2.03·13-s + 1.66·15-s − 0.805·17-s − 0.341·19-s − 1.66·21-s − 7.81·23-s + 25-s − 5.38·27-s + 6.70·29-s − 6.59·31-s + 4.32·33-s − 35-s − 11.6·37-s − 3.38·39-s − 7.40·41-s − 43-s − 0.236·45-s + 1.64·47-s + 49-s − 1.33·51-s − 10.6·53-s + 2.60·55-s + ⋯ |
| L(s) = 1 | + 0.959·3-s + 0.447·5-s − 0.377·7-s − 0.0788·9-s + 0.784·11-s − 0.564·13-s + 0.429·15-s − 0.195·17-s − 0.0783·19-s − 0.362·21-s − 1.62·23-s + 0.200·25-s − 1.03·27-s + 1.24·29-s − 1.18·31-s + 0.752·33-s − 0.169·35-s − 1.90·37-s − 0.541·39-s − 1.15·41-s − 0.152·43-s − 0.0352·45-s + 0.239·47-s + 0.142·49-s − 0.187·51-s − 1.46·53-s + 0.350·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 1.66T + 3T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 13 | \( 1 + 2.03T + 13T^{2} \) |
| 17 | \( 1 + 0.805T + 17T^{2} \) |
| 19 | \( 1 + 0.341T + 19T^{2} \) |
| 23 | \( 1 + 7.81T + 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + 6.59T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 + 7.40T + 41T^{2} \) |
| 47 | \( 1 - 1.64T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 3.05T + 59T^{2} \) |
| 61 | \( 1 - 2.07T + 61T^{2} \) |
| 67 | \( 1 + 3.60T + 67T^{2} \) |
| 71 | \( 1 - 4.96T + 71T^{2} \) |
| 73 | \( 1 + 1.91T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 4.22T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 - 7.06T + 97T^{2} \) |
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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
−7.84059769128442133725877707796, −7.00862076042901821583568922104, −6.40063009526811939081564186044, −5.63765056370897352891230837667, −4.78139670263802270947637311695, −3.79519599260801745405900838105, −3.26395883208215304118435228812, −2.29220378852288506750110604838, −1.66285570239662615042578801863, 0,
1.66285570239662615042578801863, 2.29220378852288506750110604838, 3.26395883208215304118435228812, 3.79519599260801745405900838105, 4.78139670263802270947637311695, 5.63765056370897352891230837667, 6.40063009526811939081564186044, 7.00862076042901821583568922104, 7.84059769128442133725877707796
