| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s − 3·9-s + 2·13-s − 2·14-s + 16-s − 6·17-s − 3·18-s − 6·19-s + 4·23-s + 2·26-s − 2·28-s − 4·31-s + 32-s − 6·34-s − 3·36-s − 37-s − 6·38-s − 10·41-s − 4·43-s + 4·46-s − 2·47-s − 3·49-s + 2·52-s + 2·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s − 9-s + 0.554·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.707·18-s − 1.37·19-s + 0.834·23-s + 0.392·26-s − 0.377·28-s − 0.718·31-s + 0.176·32-s − 1.02·34-s − 1/2·36-s − 0.164·37-s − 0.973·38-s − 1.56·41-s − 0.609·43-s + 0.589·46-s − 0.291·47-s − 3/7·49-s + 0.277·52-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.750324807682882988874725608634, −8.221535676991206294217792387889, −6.80613179515022113845924157121, −6.54855209691102148210096591792, −5.63131455176504702320472538865, −4.74788570792648033703597383465, −3.76886823851024934055265227029, −2.96349120039019999776178816015, −1.96778221189222956640787640137, 0,
1.96778221189222956640787640137, 2.96349120039019999776178816015, 3.76886823851024934055265227029, 4.74788570792648033703597383465, 5.63131455176504702320472538865, 6.54855209691102148210096591792, 6.80613179515022113845924157121, 8.221535676991206294217792387889, 8.750324807682882988874725608634
