L(s) = 1 | + 2-s + 4-s + 2·5-s − 7-s + 8-s + 2·10-s − 4·13-s − 14-s + 16-s − 17-s + 6·19-s + 2·20-s − 25-s − 4·26-s − 28-s − 6·29-s − 8·31-s + 32-s − 34-s − 2·35-s − 2·37-s + 6·38-s + 2·40-s + 2·41-s − 10·43-s − 2·47-s + 49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s + 0.632·10-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 1.37·19-s + 0.447·20-s − 1/5·25-s − 0.784·26-s − 0.188·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.171·34-s − 0.338·35-s − 0.328·37-s + 0.973·38-s + 0.316·40-s + 0.312·41-s − 1.52·43-s − 0.291·47-s + 1/7·49-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
\(L(1)\) |
\(\approx\) |
\(2.668025807\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.668025807\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.96595716737510, −12.44950224362301, −11.91142713672209, −11.57343536820973, −11.06602699966756, −10.49788122609233, −9.942320041157938, −9.674908367571487, −9.284145784030114, −8.738462549268289, −7.990011891859622, −7.448566256011498, −7.162703286579884, −6.606870760501561, −6.090498327094346, −5.476309564518214, −5.257406651697743, −4.846886409885890, −3.898977071445617, −3.687367261688957, −2.896197025515197, −2.524739180493840, −1.795224188899940, −1.469830773828069, −0.3562020965003408,
0.3562020965003408, 1.469830773828069, 1.795224188899940, 2.524739180493840, 2.896197025515197, 3.687367261688957, 3.898977071445617, 4.846886409885890, 5.257406651697743, 5.476309564518214, 6.090498327094346, 6.606870760501561, 7.162703286579884, 7.448566256011498, 7.990011891859622, 8.738462549268289, 9.284145784030114, 9.674908367571487, 9.942320041157938, 10.49788122609233, 11.06602699966756, 11.57343536820973, 11.91142713672209, 12.44950224362301, 12.96595716737510