| L(s) = 1 | − 2·2-s + 10·3-s − 20·6-s − 28·7-s + 8·8-s + 27·9-s − 53·11-s − 50·13-s + 56·14-s − 16·16-s + 14·17-s − 54·18-s + 95·19-s − 280·21-s + 106·22-s + 23-s + 80·24-s + 100·26-s − 190·27-s − 412·29-s − 108·31-s − 530·33-s − 28·34-s − 57·37-s − 190·38-s − 500·39-s + 486·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.92·3-s − 1.36·6-s − 1.51·7-s + 0.353·8-s + 9-s − 1.45·11-s − 1.06·13-s + 1.06·14-s − 1/4·16-s + 0.199·17-s − 0.707·18-s + 1.14·19-s − 2.90·21-s + 1.02·22-s + 0.00906·23-s + 0.680·24-s + 0.754·26-s − 1.35·27-s − 2.63·29-s − 0.625·31-s − 2.79·33-s − 0.141·34-s − 0.253·37-s − 0.811·38-s − 2.05·39-s + 1.85·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.01042948199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01042948199\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 10 T + 73 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 53 T + 1478 T^{2} + 53 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 25 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 14 T - 4717 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 p T + 6 p^{2} T^{2} - 5 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T - 12166 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 206 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 108 T - 18127 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 57 T - 47404 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 243 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 434 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 231 T - 50462 T^{2} + 231 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 263 T - 79708 T^{2} - 263 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 24 T - 204803 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 116 T - 213525 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 204 T - 259147 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 484 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 692 T + 89847 T^{2} + 692 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 466 T - 275883 T^{2} + 466 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 228 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 362 T - 573925 T^{2} - 362 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 854 T + p^{3} T^{2} )^{2} \) |
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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
−11.62653449852212042446102139838, −10.59944741008033365770995609372, −9.980884912876056519736665453697, −9.716601862681214091843336984543, −9.368383000616132148751032634179, −9.192871351758528600865688635038, −8.367380980222462789646690778089, −8.231485080810686994598404254350, −7.52470400170035615181889654244, −7.38677102647184090540279533039, −6.87663034429098675884645744904, −5.81600889575501099368160945527, −5.52145120538479514124072895137, −4.86069098557007738511345919304, −3.70778416748934452606666920858, −3.49916583501781047638451783395, −2.80864797526560914737190981383, −2.49470828326172294067253878203, −1.68917098489123767123829062158, −0.03314158824316176183718359528,
0.03314158824316176183718359528, 1.68917098489123767123829062158, 2.49470828326172294067253878203, 2.80864797526560914737190981383, 3.49916583501781047638451783395, 3.70778416748934452606666920858, 4.86069098557007738511345919304, 5.52145120538479514124072895137, 5.81600889575501099368160945527, 6.87663034429098675884645744904, 7.38677102647184090540279533039, 7.52470400170035615181889654244, 8.231485080810686994598404254350, 8.367380980222462789646690778089, 9.192871351758528600865688635038, 9.368383000616132148751032634179, 9.716601862681214091843336984543, 9.980884912876056519736665453697, 10.59944741008033365770995609372, 11.62653449852212042446102139838
