| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 4·6-s − 4·8-s + 3·9-s + 2·11-s + 6·12-s − 4·13-s + 5·16-s − 8·17-s − 6·18-s − 2·19-s − 4·22-s − 2·23-s − 8·24-s + 8·26-s + 4·27-s − 14·29-s − 2·31-s − 6·32-s + 4·33-s + 16·34-s + 9·36-s − 20·37-s + 4·38-s − 8·39-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s − 1.41·8-s + 9-s + 0.603·11-s + 1.73·12-s − 1.10·13-s + 5/4·16-s − 1.94·17-s − 1.41·18-s − 0.458·19-s − 0.852·22-s − 0.417·23-s − 1.63·24-s + 1.56·26-s + 0.769·27-s − 2.59·29-s − 0.359·31-s − 1.06·32-s + 0.696·33-s + 2.74·34-s + 3/2·36-s − 3.28·37-s + 0.648·38-s − 1.28·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 14 T + 97 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 23 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 112 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 121 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 113 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 136 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 119 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 139 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 20 T + 204 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
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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
−8.601066503512158468956862718030, −8.487150237271871932613429286926, −7.74741177340978697059668090317, −7.66345948291603500129618933337, −7.26430638784424561037313504039, −6.77958575296109261392586491330, −6.65591232676171535683224167358, −6.22547768805342446486454866887, −5.38391687667509678727337298964, −5.33008398925280759863601443405, −4.38928191573525178942260395744, −4.29604835594538662977575648789, −3.53545487256366234183197361891, −3.33188453095709915792468874551, −2.63063946994504521055797895436, −2.09778565179684852817163246333, −1.86256759899833465412097939838, −1.48473002893273732988420711179, 0, 0,
1.48473002893273732988420711179, 1.86256759899833465412097939838, 2.09778565179684852817163246333, 2.63063946994504521055797895436, 3.33188453095709915792468874551, 3.53545487256366234183197361891, 4.29604835594538662977575648789, 4.38928191573525178942260395744, 5.33008398925280759863601443405, 5.38391687667509678727337298964, 6.22547768805342446486454866887, 6.65591232676171535683224167358, 6.77958575296109261392586491330, 7.26430638784424561037313504039, 7.66345948291603500129618933337, 7.74741177340978697059668090317, 8.487150237271871932613429286926, 8.601066503512158468956862718030
