| L(s) = 1 | + 1.41·2-s − 3-s + 1.00·4-s − 1.41·6-s + 9-s − 1.41·11-s − 1.00·12-s − 0.999·16-s − 1.41·17-s + 1.41·18-s − 2.00·22-s + 1.41·23-s + 25-s − 27-s + 1.41·29-s − 1.41·32-s + 1.41·33-s − 2.00·34-s + 1.00·36-s − 1.41·44-s + 2.00·46-s + 0.999·48-s + 49-s + 1.41·50-s + 1.41·51-s − 1.41·53-s − 1.41·54-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 3-s + 1.00·4-s − 1.41·6-s + 9-s − 1.41·11-s − 1.00·12-s − 0.999·16-s − 1.41·17-s + 1.41·18-s − 2.00·22-s + 1.41·23-s + 25-s − 27-s + 1.41·29-s − 1.41·32-s + 1.41·33-s − 2.00·34-s + 1.00·36-s − 1.41·44-s + 2.00·46-s + 0.999·48-s + 49-s + 1.41·50-s + 1.41·51-s − 1.41·53-s − 1.41·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9399289817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9399289817\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 1.41T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.41T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + 1.41T + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 - 1.41T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.41T + T^{2} \) |
| 59 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 + 2T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 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
−13.03631595072602093843039961165, −12.11953725416449572094281884343, −11.11795954886805531459029265260, −10.43925115117384294982861425918, −8.857296015536400237529541299715, −7.18646828509669977662916339740, −6.26991446017201404893330014309, −5.12579395998211793256364559476, −4.53250682818961780933031255608, −2.79385169794189637840328832684,
2.79385169794189637840328832684, 4.53250682818961780933031255608, 5.12579395998211793256364559476, 6.26991446017201404893330014309, 7.18646828509669977662916339740, 8.857296015536400237529541299715, 10.43925115117384294982861425918, 11.11795954886805531459029265260, 12.11953725416449572094281884343, 13.03631595072602093843039961165
