| L(s) = 1 | − 2·2-s + 2·4-s − 5-s + 3·7-s − 4·8-s + 2·10-s − 2·11-s + 5·13-s − 6·14-s + 8·16-s + 16·17-s + 2·19-s − 2·20-s + 4·22-s + 6·23-s − 10·26-s + 6·28-s + 2·29-s − 8·32-s − 32·34-s − 3·35-s + 10·37-s − 4·38-s + 4·40-s − 10·41-s − 4·43-s − 4·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s + 1.13·7-s − 1.41·8-s + 0.632·10-s − 0.603·11-s + 1.38·13-s − 1.60·14-s + 2·16-s + 3.88·17-s + 0.458·19-s − 0.447·20-s + 0.852·22-s + 1.25·23-s − 1.96·26-s + 1.13·28-s + 0.371·29-s − 1.41·32-s − 5.48·34-s − 0.507·35-s + 1.64·37-s − 0.648·38-s + 0.632·40-s − 1.56·41-s − 0.609·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9669490097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9669490097\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−11.43889871059080265674462485987, −10.88453421847762944602818442222, −10.40718919084505880155866124730, −10.21935532222784168167537847357, −9.437266625424820203166980756257, −9.399007788310623279621926700885, −8.486703952539787754181347501365, −8.363861304114176456682390273130, −7.916929550795261082861775466940, −7.58408491165739626257950173207, −7.18331000262169702589735157948, −6.14692839092709099640451756561, −5.91061328738847037278892608523, −5.25076995185891532626547408924, −4.89241394723324375503901754068, −3.62444156160475202883747639614, −3.38933942126719167586635050708, −2.71398954237525408988619170929, −1.22103081696222375908404614357, −1.08343691321948729009881248607,
1.08343691321948729009881248607, 1.22103081696222375908404614357, 2.71398954237525408988619170929, 3.38933942126719167586635050708, 3.62444156160475202883747639614, 4.89241394723324375503901754068, 5.25076995185891532626547408924, 5.91061328738847037278892608523, 6.14692839092709099640451756561, 7.18331000262169702589735157948, 7.58408491165739626257950173207, 7.916929550795261082861775466940, 8.363861304114176456682390273130, 8.486703952539787754181347501365, 9.399007788310623279621926700885, 9.437266625424820203166980756257, 10.21935532222784168167537847357, 10.40718919084505880155866124730, 10.88453421847762944602818442222, 11.43889871059080265674462485987
