| L(s) = 1 | − 9-s − 8·11-s + 8·19-s + 20·29-s − 12·41-s − 49-s + 8·59-s − 20·61-s − 32·71-s − 16·79-s + 81-s − 20·89-s + 8·99-s − 20·101-s − 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s − 8·171-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 2.41·11-s + 1.83·19-s + 3.71·29-s − 1.87·41-s − 1/7·49-s + 1.04·59-s − 2.56·61-s − 3.79·71-s − 1.80·79-s + 1/9·81-s − 2.11·89-s + 0.804·99-s − 1.99·101-s − 2.68·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s − 0.611·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6885374873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6885374873\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.521443766495321936964338769319, −8.200643087731278099040476949921, −7.934549514660148073035207717774, −7.52014669385948410308618551787, −7.23556044699650181363597299832, −6.68552741638615840665030529596, −6.57040054913897829600979146721, −5.80000228168391721832248114875, −5.61568030644377643100687478575, −5.26383555024728082949339658230, −4.93738674114803026501702692710, −4.45451269102076536026175052377, −4.27963955700977926189438003349, −3.29203438297566597302190092751, −3.02631142049500790593921527395, −2.73920277504955962296386464521, −2.58728416385537373803258724699, −1.47078435540428594792114919328, −1.27127149008957950654286886967, −0.24135548249415434020466521256,
0.24135548249415434020466521256, 1.27127149008957950654286886967, 1.47078435540428594792114919328, 2.58728416385537373803258724699, 2.73920277504955962296386464521, 3.02631142049500790593921527395, 3.29203438297566597302190092751, 4.27963955700977926189438003349, 4.45451269102076536026175052377, 4.93738674114803026501702692710, 5.26383555024728082949339658230, 5.61568030644377643100687478575, 5.80000228168391721832248114875, 6.57040054913897829600979146721, 6.68552741638615840665030529596, 7.23556044699650181363597299832, 7.52014669385948410308618551787, 7.934549514660148073035207717774, 8.200643087731278099040476949921, 8.521443766495321936964338769319
