| L(s) = 1 | − 3-s + 5-s + 3·9-s + 5·11-s + 14·13-s − 15-s + 3·17-s + 2·19-s − 8·23-s − 8·27-s − 10·29-s + 10·31-s − 5·33-s − 4·37-s − 14·39-s − 12·41-s + 4·43-s + 3·45-s + 7·47-s − 3·51-s + 10·53-s + 5·55-s − 2·57-s + 10·59-s + 12·61-s + 14·65-s + 2·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 9-s + 1.50·11-s + 3.88·13-s − 0.258·15-s + 0.727·17-s + 0.458·19-s − 1.66·23-s − 1.53·27-s − 1.85·29-s + 1.79·31-s − 0.870·33-s − 0.657·37-s − 2.24·39-s − 1.87·41-s + 0.609·43-s + 0.447·45-s + 1.02·47-s − 0.420·51-s + 1.37·53-s + 0.674·55-s − 0.264·57-s + 1.30·59-s + 1.53·61-s + 1.73·65-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.806106770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.806106770\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 8 T - 25 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
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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
−9.207862092362180604875181255558, −9.126119317161688301133582284278, −8.474512193371282102391668971108, −8.431537647859188702113352040082, −7.79349300977547411821340285375, −7.47100262669678537093412087801, −6.73782105966225032770389192735, −6.55574679815644722996309456902, −6.10032973042187548627168270274, −6.03816549072262848861062689395, −5.32281802156415693364902181508, −5.28762446757004117459187756400, −4.07160894204032807496432165305, −4.01488478793836848553614459490, −3.62581920886289565601230883121, −3.53349353690024445423305834036, −2.26680906458226102063981141964, −1.71905210619224407173598310195, −1.23121649295284001054239567466, −0.901026727314195518823323919882,
0.901026727314195518823323919882, 1.23121649295284001054239567466, 1.71905210619224407173598310195, 2.26680906458226102063981141964, 3.53349353690024445423305834036, 3.62581920886289565601230883121, 4.01488478793836848553614459490, 4.07160894204032807496432165305, 5.28762446757004117459187756400, 5.32281802156415693364902181508, 6.03816549072262848861062689395, 6.10032973042187548627168270274, 6.55574679815644722996309456902, 6.73782105966225032770389192735, 7.47100262669678537093412087801, 7.79349300977547411821340285375, 8.431537647859188702113352040082, 8.474512193371282102391668971108, 9.126119317161688301133582284278, 9.207862092362180604875181255558
