| L(s) = 1 | − 2·2-s + 3·4-s − 4·7-s − 4·8-s + 4·11-s + 8·14-s + 5·16-s − 8·22-s + 6·23-s − 8·25-s − 12·28-s − 8·29-s − 6·32-s − 4·37-s + 2·43-s + 12·44-s − 12·46-s + 9·49-s + 16·50-s + 26·53-s + 16·56-s + 16·58-s + 7·64-s + 8·67-s + 2·71-s + 8·74-s − 16·77-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.51·7-s − 1.41·8-s + 1.20·11-s + 2.13·14-s + 5/4·16-s − 1.70·22-s + 1.25·23-s − 8/5·25-s − 2.26·28-s − 1.48·29-s − 1.06·32-s − 0.657·37-s + 0.304·43-s + 1.80·44-s − 1.76·46-s + 9/7·49-s + 2.26·50-s + 3.57·53-s + 2.13·56-s + 2.10·58-s + 7/8·64-s + 0.977·67-s + 0.237·71-s + 0.929·74-s − 1.82·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
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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
−7.46279151861437526314386022165, −7.15761375511992423287681446444, −6.83249047217393470688211331264, −6.56263511591907701222430269084, −5.93086545754445054792998745813, −5.66203974026568150858358116795, −5.22018101456839073850602135358, −4.20496168249042371040397667236, −3.67930606619282856977205375106, −3.63066911118268614071863051332, −2.73402706483716872013713425401, −2.32420571079081891891836612602, −1.58128716775591137241953666416, −0.864265341674288841465502441553, 0,
0.864265341674288841465502441553, 1.58128716775591137241953666416, 2.32420571079081891891836612602, 2.73402706483716872013713425401, 3.63066911118268614071863051332, 3.67930606619282856977205375106, 4.20496168249042371040397667236, 5.22018101456839073850602135358, 5.66203974026568150858358116795, 5.93086545754445054792998745813, 6.56263511591907701222430269084, 6.83249047217393470688211331264, 7.15761375511992423287681446444, 7.46279151861437526314386022165
