| L(s) = 1 | + 2·3-s + 3·9-s + 2·11-s + 2·19-s + 2·25-s + 4·27-s + 4·33-s + 16·41-s + 8·43-s + 8·49-s + 4·57-s + 6·59-s + 2·67-s − 16·73-s + 4·75-s + 5·81-s − 14·83-s − 8·89-s + 10·97-s + 6·99-s + 10·107-s + 22·113-s − 18·121-s + 32·123-s + 127-s + 16·129-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 0.603·11-s + 0.458·19-s + 2/5·25-s + 0.769·27-s + 0.696·33-s + 2.49·41-s + 1.21·43-s + 8/7·49-s + 0.529·57-s + 0.781·59-s + 0.244·67-s − 1.87·73-s + 0.461·75-s + 5/9·81-s − 1.53·83-s − 0.847·89-s + 1.01·97-s + 0.603·99-s + 0.966·107-s + 2.06·113-s − 1.63·121-s + 2.88·123-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.395364506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.395364506\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 92 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 120 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−7.48757105581154396982704267068, −7.34108022557360245982827824347, −7.03371746228626538586327801283, −6.31425343709209009466256230011, −5.99755060510693243368471852168, −5.57484488866835796415364596524, −4.96712384199268878517889808169, −4.38566054670617352495351460147, −4.08713442706128697543072519622, −3.70763158483888691624838640998, −3.00714163035746205747536711498, −2.66012256318440546106363711583, −2.18024168985490444293073092565, −1.38391920927294270514563223916, −0.820874659491479639983897161037,
0.820874659491479639983897161037, 1.38391920927294270514563223916, 2.18024168985490444293073092565, 2.66012256318440546106363711583, 3.00714163035746205747536711498, 3.70763158483888691624838640998, 4.08713442706128697543072519622, 4.38566054670617352495351460147, 4.96712384199268878517889808169, 5.57484488866835796415364596524, 5.99755060510693243368471852168, 6.31425343709209009466256230011, 7.03371746228626538586327801283, 7.34108022557360245982827824347, 7.48757105581154396982704267068
