| L(s) = 1 | − 4·9-s + 2·11-s + 8·23-s − 10·25-s + 4·29-s − 8·37-s − 4·43-s − 8·53-s + 16·67-s + 20·79-s + 7·81-s − 8·99-s + 24·107-s − 20·109-s + 28·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 24·169-s + 173-s + ⋯ |
| L(s) = 1 | − 4/3·9-s + 0.603·11-s + 1.66·23-s − 2·25-s + 0.742·29-s − 1.31·37-s − 0.609·43-s − 1.09·53-s + 1.95·67-s + 2.25·79-s + 7/9·81-s − 0.804·99-s + 2.32·107-s − 1.91·109-s + 2.63·113-s + 3/11·121-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 − 1.84·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.140752530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.140752530\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.77773029194766291346397521597, −7.76532349783464844891761850387, −7.31570795958605675196121059154, −6.76857912315226457974027880006, −6.48012704470029264483821016374, −6.42804071872238567545725638699, −5.81016027150786753030176546483, −5.55189401736071987527346483587, −5.21003066326205883340873652923, −4.81831397513320104166654942614, −4.59860630989687139014315122487, −3.89289789310361993324502027271, −3.50161146460124686265268334419, −3.49639912544580432650751806524, −2.76688555823073277612884725666, −2.59177309663900004912697518230, −1.81859562679638357524899357399, −1.72786005863813718184468460179, −0.830324448467241561206188419656, −0.41802710742323758167636495180,
0.41802710742323758167636495180, 0.830324448467241561206188419656, 1.72786005863813718184468460179, 1.81859562679638357524899357399, 2.59177309663900004912697518230, 2.76688555823073277612884725666, 3.49639912544580432650751806524, 3.50161146460124686265268334419, 3.89289789310361993324502027271, 4.59860630989687139014315122487, 4.81831397513320104166654942614, 5.21003066326205883340873652923, 5.55189401736071987527346483587, 5.81016027150786753030176546483, 6.42804071872238567545725638699, 6.48012704470029264483821016374, 6.76857912315226457974027880006, 7.31570795958605675196121059154, 7.76532349783464844891761850387, 7.77773029194766291346397521597
