| L(s) = 1 | − 2-s + 4-s + 4·5-s + 7-s − 8-s − 4·10-s + 2·13-s − 14-s + 16-s + 17-s − 7·19-s + 4·20-s + 2·23-s + 11·25-s − 2·26-s + 28-s − 2·29-s − 32-s − 34-s + 4·35-s + 10·37-s + 7·38-s − 4·40-s − 12·41-s + 4·43-s − 2·46-s − 2·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.377·7-s − 0.353·8-s − 1.26·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 1.60·19-s + 0.894·20-s + 0.417·23-s + 11/5·25-s − 0.392·26-s + 0.188·28-s − 0.371·29-s − 0.176·32-s − 0.171·34-s + 0.676·35-s + 1.64·37-s + 1.13·38-s − 0.632·40-s − 1.87·41-s + 0.609·43-s − 0.294·46-s − 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.97495008541169, −12.74001811400126, −12.17467650316091, −11.47273932285083, −11.00583195423385, −10.63338578023607, −10.34245609572630, −9.669625044040723, −9.433187696567359, −8.909181604963245, −8.544972495073146, −7.983012718922309, −7.515917884412755, −6.756744911509455, −6.373208264319434, −6.054477787566383, −5.640058078138344, −4.789365407015513, −4.669786060939352, −3.680847485665222, −3.050410505043276, −2.485899125200422, −1.899690524080537, −1.572814627721174, −0.9573719991621021, 0,
0.9573719991621021, 1.572814627721174, 1.899690524080537, 2.485899125200422, 3.050410505043276, 3.680847485665222, 4.669786060939352, 4.789365407015513, 5.640058078138344, 6.054477787566383, 6.373208264319434, 6.756744911509455, 7.515917884412755, 7.983012718922309, 8.544972495073146, 8.909181604963245, 9.433187696567359, 9.669625044040723, 10.34245609572630, 10.63338578023607, 11.00583195423385, 11.47273932285083, 12.17467650316091, 12.74001811400126, 12.97495008541169
