| L(s) = 1 | + 1.24·2-s + 0.554·4-s + 7-s − 0.554·8-s + 9-s − 0.445·11-s + 1.24·14-s − 1.24·16-s + 1.24·18-s − 0.554·22-s − 1.80·23-s + 25-s + 0.554·28-s + 1.24·29-s − 0.999·32-s + 0.554·36-s − 1.80·37-s + 1.24·43-s − 0.246·44-s − 2.24·46-s + 49-s + 1.24·50-s − 1.80·53-s − 0.554·56-s + 1.55·58-s + 63-s − 1.80·67-s + ⋯ |
| L(s) = 1 | + 1.24·2-s + 0.554·4-s + 7-s − 0.554·8-s + 9-s − 0.445·11-s + 1.24·14-s − 1.24·16-s + 1.24·18-s − 0.554·22-s − 1.80·23-s + 25-s + 0.554·28-s + 1.24·29-s − 0.999·32-s + 0.554·36-s − 1.80·37-s + 1.24·43-s − 0.246·44-s − 2.24·46-s + 49-s + 1.24·50-s − 1.80·53-s − 0.554·56-s + 1.55·58-s + 63-s − 1.80·67-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.002260863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.002260863\) |
| \(L(1)\) |
|
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 | \( 1 - 1.24T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + 0.445T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.80T + T^{2} \) |
| 29 | \( 1 - 1.24T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.80T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.24T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.80T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.80T + T^{2} \) |
| 71 | \( 1 + 1.80T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 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
−10.21519293390444610120439714360, −9.092476460900286484826854097985, −8.229566456355808131590381312746, −7.36567232140555998241399662143, −6.42642045130246117957607012423, −5.51038466448409364761063797877, −4.65747307404811651378247993182, −4.17982861208637619589798514224, −2.96958466160149285251715099321, −1.73653318792125822542248090049,
1.73653318792125822542248090049, 2.96958466160149285251715099321, 4.17982861208637619589798514224, 4.65747307404811651378247993182, 5.51038466448409364761063797877, 6.42642045130246117957607012423, 7.36567232140555998241399662143, 8.229566456355808131590381312746, 9.092476460900286484826854097985, 10.21519293390444610120439714360
