| L(s) = 1 | + 8·3-s + 8·7-s + 18·9-s − 64·11-s − 12·13-s − 4·17-s − 208·19-s + 64·21-s + 120·23-s − 8·27-s − 292·29-s − 176·31-s − 512·33-s + 356·37-s − 96·39-s + 100·41-s − 376·43-s − 280·47-s − 38·49-s − 32·51-s − 316·53-s − 1.66e3·57-s − 720·59-s − 1.26e3·61-s + 144·63-s − 744·67-s + 960·69-s + ⋯ |
| L(s) = 1 | + 1.53·3-s + 0.431·7-s + 2/3·9-s − 1.75·11-s − 0.256·13-s − 0.0570·17-s − 2.51·19-s + 0.665·21-s + 1.08·23-s − 0.0570·27-s − 1.86·29-s − 1.01·31-s − 2.70·33-s + 1.58·37-s − 0.394·39-s + 0.380·41-s − 1.33·43-s − 0.868·47-s − 0.110·49-s − 0.0878·51-s − 0.818·53-s − 3.86·57-s − 1.58·59-s − 2.66·61-s + 0.287·63-s − 1.35·67-s + 1.67·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 64 T + 2822 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 12 T + 2894 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T - 3994 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 208 T + 22998 T^{2} + 208 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 120 T + 25030 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 292 T + 63950 T^{2} + 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 176 T + 66462 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 356 T + 126846 T^{2} - 356 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 100 T + 101942 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 376 T + 161502 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 280 T + 223190 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 316 T + 308894 T^{2} + 316 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 720 T + 530758 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1268 T + 831342 T^{2} + 1268 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 744 T + 686894 T^{2} + 744 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 48 T - 424178 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 940 T + 653334 T^{2} - 940 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 32 T + 276318 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1592 T + 1724174 T^{2} + 1592 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 780 T + 1506742 T^{2} + 780 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1220 T + 2159046 T^{2} + 1220 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.632716843496427546404572400523, −9.148878329580374857786454891748, −8.593304621119024073442427604903, −8.557058803411616523961723471767, −7.955014332681702620008394762995, −7.69305835362538123058926320177, −7.34547720244630895624301375862, −6.75278910198965300806696696118, −5.94001357815934568666757928753, −5.90281291958114340082665493799, −4.87358611295253169730677890300, −4.78698759715157281035192874643, −4.16204690013946083972675432103, −3.44170017692573098830153285907, −2.94568965488435730607855121222, −2.61249773450209371304956926580, −1.96897793695259334757241162074, −1.59248845159571050254581453507, 0, 0,
1.59248845159571050254581453507, 1.96897793695259334757241162074, 2.61249773450209371304956926580, 2.94568965488435730607855121222, 3.44170017692573098830153285907, 4.16204690013946083972675432103, 4.78698759715157281035192874643, 4.87358611295253169730677890300, 5.90281291958114340082665493799, 5.94001357815934568666757928753, 6.75278910198965300806696696118, 7.34547720244630895624301375862, 7.69305835362538123058926320177, 7.955014332681702620008394762995, 8.557058803411616523961723471767, 8.593304621119024073442427604903, 9.148878329580374857786454891748, 9.632716843496427546404572400523
