| L(s) = 1 | − 1.80·3-s − 5-s + 1.24·7-s + 2.24·9-s + 0.445·11-s + 1.80·15-s − 2.24·21-s + 25-s − 2.24·27-s − 1.24·31-s − 0.801·33-s − 1.24·35-s + 0.445·37-s − 1.80·41-s + 43-s − 2.24·45-s + 0.554·49-s − 0.445·55-s + 1.80·59-s + 2.80·63-s + 1.80·73-s − 1.80·75-s + 0.554·77-s + 0.445·79-s + 1.80·81-s + 2.24·93-s + 99-s + ⋯ |
| L(s) = 1 | − 1.80·3-s − 5-s + 1.24·7-s + 2.24·9-s + 0.445·11-s + 1.80·15-s − 2.24·21-s + 25-s − 2.24·27-s − 1.24·31-s − 0.801·33-s − 1.24·35-s + 0.445·37-s − 1.80·41-s + 43-s − 2.24·45-s + 0.554·49-s − 0.445·55-s + 1.80·59-s + 2.80·63-s + 1.80·73-s − 1.80·75-s + 0.554·77-s + 0.445·79-s + 1.80·81-s + 2.24·93-s + 99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6457373832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6457373832\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + 1.80T + T^{2} \) |
| 7 | \( 1 - 1.24T + T^{2} \) |
| 11 | \( 1 - 0.445T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.24T + T^{2} \) |
| 37 | \( 1 - 0.445T + T^{2} \) |
| 41 | \( 1 + 1.80T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.80T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.80T + T^{2} \) |
| 79 | \( 1 - 0.445T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.632676044294980042462778108988, −7.87536422970996377564703514695, −7.16445325541494103558513809931, −6.59090455210949658555848215373, −5.58973859155163287801339255573, −5.03174172455578800406646774494, −4.37714525939965876373810290205, −3.63958299228820204685109544033, −1.86687964995406216574829873868, −0.799178317613072066596658265775,
0.799178317613072066596658265775, 1.86687964995406216574829873868, 3.63958299228820204685109544033, 4.37714525939965876373810290205, 5.03174172455578800406646774494, 5.58973859155163287801339255573, 6.59090455210949658555848215373, 7.16445325541494103558513809931, 7.87536422970996377564703514695, 8.632676044294980042462778108988
