| L(s) = 1 | + 2·2-s + 3·4-s + 2·7-s + 4·8-s + 4·11-s − 2·13-s + 4·14-s + 5·16-s − 10·17-s − 8·19-s + 8·22-s − 6·23-s − 4·26-s + 6·28-s + 2·29-s + 2·31-s + 6·32-s − 20·34-s − 12·37-s − 16·38-s − 8·41-s − 14·43-s + 12·44-s − 12·46-s + 4·47-s + 3·49-s − 6·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.755·7-s + 1.41·8-s + 1.20·11-s − 0.554·13-s + 1.06·14-s + 5/4·16-s − 2.42·17-s − 1.83·19-s + 1.70·22-s − 1.25·23-s − 0.784·26-s + 1.13·28-s + 0.371·29-s + 0.359·31-s + 1.06·32-s − 3.42·34-s − 1.97·37-s − 2.59·38-s − 1.24·41-s − 2.13·43-s + 1.80·44-s − 1.76·46-s + 0.583·47-s + 3/7·49-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 17 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 44 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 45 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 86 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 18 T + 175 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 61 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 20 T + 248 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 130 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 219 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 188 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.24123915431750675823216960011, −7.08323139593088911593461231996, −6.65413419083383114114096329771, −6.45435504146751086895342964227, −6.28827778407599451143684437855, −5.86737116452966091414934787619, −5.29635600342569658601727429274, −5.08474127521247381947592990987, −4.55867102477661273753914422532, −4.39827583339048642594067271675, −4.17907284765987261173876221227, −3.89595472700994267998189889485, −3.16029558051035523207604603660, −3.06405774165612595731684759890, −2.19045379755382407101031136167, −2.18256989492587604085649459610, −1.55362884393722454543414939829, −1.52907631095352486749767915256, 0, 0,
1.52907631095352486749767915256, 1.55362884393722454543414939829, 2.18256989492587604085649459610, 2.19045379755382407101031136167, 3.06405774165612595731684759890, 3.16029558051035523207604603660, 3.89595472700994267998189889485, 4.17907284765987261173876221227, 4.39827583339048642594067271675, 4.55867102477661273753914422532, 5.08474127521247381947592990987, 5.29635600342569658601727429274, 5.86737116452966091414934787619, 6.28827778407599451143684437855, 6.45435504146751086895342964227, 6.65413419083383114114096329771, 7.08323139593088911593461231996, 7.24123915431750675823216960011
