| L(s) = 1 | + 3-s − 6·5-s − 7-s + 6·11-s − 6·15-s − 12·17-s + 7·19-s − 21-s + 19·25-s − 27-s + 5·31-s + 6·33-s + 6·35-s − 37-s − 6·47-s − 6·49-s − 12·51-s − 36·55-s + 7·57-s − 3·67-s + 15·73-s + 19·75-s − 6·77-s + 27·79-s − 81-s + 12·83-s + 72·85-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2.68·5-s − 0.377·7-s + 1.80·11-s − 1.54·15-s − 2.91·17-s + 1.60·19-s − 0.218·21-s + 19/5·25-s − 0.192·27-s + 0.898·31-s + 1.04·33-s + 1.01·35-s − 0.164·37-s − 0.875·47-s − 6/7·49-s − 1.68·51-s − 4.85·55-s + 0.927·57-s − 0.366·67-s + 1.75·73-s + 2.19·75-s − 0.683·77-s + 3.03·79-s − 1/9·81-s + 1.31·83-s + 7.80·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9151349854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9151349854\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 15 T + 148 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 27 T + 322 T^{2} - 27 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−11.67343611103636732705167589079, −11.38847337403904276789109364749, −11.15567492850527197158541605402, −10.64139031036348221694708995806, −9.715169803852317281181358380869, −9.286997483147897225196553266053, −8.932936969838679782927063053046, −8.522587200931821108660134090475, −7.920550024531798814450007487212, −7.72256228579704899339448103983, −6.96202234320507874316890370330, −6.65100655823187627997063639449, −6.33830501305346786678490751988, −5.01217475408312362166444693423, −4.52602504239569546272025815219, −4.11232683552099316476351158618, −3.47617612560870217028195536793, −3.29089561994621060237327586837, −2.11988581949543678173896705132, −0.66595988916745166209948767685,
0.66595988916745166209948767685, 2.11988581949543678173896705132, 3.29089561994621060237327586837, 3.47617612560870217028195536793, 4.11232683552099316476351158618, 4.52602504239569546272025815219, 5.01217475408312362166444693423, 6.33830501305346786678490751988, 6.65100655823187627997063639449, 6.96202234320507874316890370330, 7.72256228579704899339448103983, 7.920550024531798814450007487212, 8.522587200931821108660134090475, 8.932936969838679782927063053046, 9.286997483147897225196553266053, 9.715169803852317281181358380869, 10.64139031036348221694708995806, 11.15567492850527197158541605402, 11.38847337403904276789109364749, 11.67343611103636732705167589079
