| L(s) = 1 | − 13-s − 17-s − 4·19-s − 6·29-s + 2·37-s − 6·41-s + 2·43-s − 4·47-s − 7·49-s − 4·53-s − 2·59-s + 8·61-s − 4·67-s − 8·71-s + 4·73-s − 10·79-s − 12·83-s − 12·89-s − 4·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.277·13-s − 0.242·17-s − 0.917·19-s − 1.11·29-s + 0.328·37-s − 0.937·41-s + 0.304·43-s − 0.583·47-s − 49-s − 0.549·53-s − 0.260·59-s + 1.02·61-s − 0.488·67-s − 0.949·71-s + 0.468·73-s − 1.12·79-s − 1.31·83-s − 1.27·89-s − 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06619146068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06619146068\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−12.64542806185460, −11.83181007611628, −11.60191179909433, −11.09199651253166, −10.67095030661543, −10.20450028150535, −9.672939662477851, −9.377587663698297, −8.759835189148955, −8.366510162192301, −7.933487590772655, −7.414947512600480, −6.814316205985842, −6.613219554075285, −5.871608831062405, −5.557965372824064, −4.939163253608751, −4.409754071738668, −4.039007781389304, −3.382733679421830, −2.869836541281083, −2.282032538152003, −1.700759132694422, −1.203908303193664, −0.06421509634437917,
0.06421509634437917, 1.203908303193664, 1.700759132694422, 2.282032538152003, 2.869836541281083, 3.382733679421830, 4.039007781389304, 4.409754071738668, 4.939163253608751, 5.557965372824064, 5.871608831062405, 6.613219554075285, 6.814316205985842, 7.414947512600480, 7.933487590772655, 8.366510162192301, 8.759835189148955, 9.377587663698297, 9.672939662477851, 10.20450028150535, 10.67095030661543, 11.09199651253166, 11.60191179909433, 11.83181007611628, 12.64542806185460
