| L(s) = 1 | − 2-s + 4-s + 1.41·7-s − 8-s + 9-s − 1.41·13-s − 1.41·14-s + 16-s − 18-s − 1.41·23-s + 25-s + 1.41·26-s + 1.41·28-s + 1.41·29-s − 1.41·31-s − 32-s + 36-s + 1.41·37-s + 1.41·46-s + 1.00·49-s − 50-s − 1.41·52-s − 1.41·56-s − 1.41·58-s − 1.41·61-s + 1.41·62-s + 1.41·63-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 1.41·7-s − 8-s + 9-s − 1.41·13-s − 1.41·14-s + 16-s − 18-s − 1.41·23-s + 25-s + 1.41·26-s + 1.41·28-s + 1.41·29-s − 1.41·31-s − 32-s + 36-s + 1.41·37-s + 1.41·46-s + 1.00·49-s − 50-s − 1.41·52-s − 1.41·56-s − 1.41·58-s − 1.41·61-s + 1.41·62-s + 1.41·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7326162615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7326162615\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 1.41T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + 1.41T + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.41T + T^{2} \) |
| 29 | \( 1 - 1.41T + T^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 - 1.41T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.41T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 97 | \( 1 + 2T + T^{2} \) |
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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
−10.46461623015912342288518243655, −9.843136176538583349311823413342, −8.930367256886645413127378852174, −7.901643825972310629940558619764, −7.51196479762056599109267913191, −6.51680832256976707900337698995, −5.20713323134544906053192825853, −4.28320141595308205219903610134, −2.53035238070173749026039412099, −1.47660825665310258020429476082,
1.47660825665310258020429476082, 2.53035238070173749026039412099, 4.28320141595308205219903610134, 5.20713323134544906053192825853, 6.51680832256976707900337698995, 7.51196479762056599109267913191, 7.901643825972310629940558619764, 8.930367256886645413127378852174, 9.843136176538583349311823413342, 10.46461623015912342288518243655
