| L(s) = 1 | + 0.445·2-s − 0.801·4-s − 7-s − 0.801·8-s + 9-s + 1.80·11-s − 0.445·14-s + 0.445·16-s + 0.445·18-s + 0.801·22-s + 1.24·23-s + 25-s + 0.801·28-s − 0.445·29-s + 32-s − 0.801·36-s − 1.24·37-s − 0.445·43-s − 1.44·44-s + 0.554·46-s + 49-s + 0.445·50-s + 1.24·53-s + 0.801·56-s − 0.198·58-s − 63-s − 1.24·67-s + ⋯ |
| L(s) = 1 | + 0.445·2-s − 0.801·4-s − 7-s − 0.801·8-s + 9-s + 1.80·11-s − 0.445·14-s + 0.445·16-s + 0.445·18-s + 0.801·22-s + 1.24·23-s + 25-s + 0.801·28-s − 0.445·29-s + 32-s − 0.801·36-s − 1.24·37-s − 0.445·43-s − 1.44·44-s + 0.554·46-s + 49-s + 0.445·50-s + 1.24·53-s + 0.801·56-s − 0.198·58-s − 63-s − 1.24·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.124105853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.124105853\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.445T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.80T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.24T + T^{2} \) |
| 29 | \( 1 + 0.445T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.24T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 0.445T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.24T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.24T + T^{2} \) |
| 71 | \( 1 + 1.24T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.80T + T^{2} \) |
| 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
−9.814884100680104285798110421047, −9.110193835879100367543651472175, −8.719719355514806014371444881622, −7.15017813320853565914871129888, −6.70362512567491693130547239584, −5.71961757164950928052100835622, −4.63226577070451262515644921150, −3.89486236525813638738291495744, −3.13726421289603727885589236212, −1.25093772264914068236712149944,
1.25093772264914068236712149944, 3.13726421289603727885589236212, 3.89486236525813638738291495744, 4.63226577070451262515644921150, 5.71961757164950928052100835622, 6.70362512567491693130547239584, 7.15017813320853565914871129888, 8.719719355514806014371444881622, 9.110193835879100367543651472175, 9.814884100680104285798110421047
