| L(s) = 1 | + 3·2-s − 6·3-s − 5·4-s − 18·6-s − 14·7-s − 33·8-s + 27·9-s − 31·11-s + 30·12-s + 39·13-s − 42·14-s − 21·16-s − 79·17-s + 81·18-s − 56·19-s + 84·21-s − 93·22-s + 254·23-s + 198·24-s + 117·26-s − 108·27-s + 70·28-s − 62·29-s − 135·31-s + 87·32-s + 186·33-s − 237·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 1.15·3-s − 5/8·4-s − 1.22·6-s − 0.755·7-s − 1.45·8-s + 9-s − 0.849·11-s + 0.721·12-s + 0.832·13-s − 0.801·14-s − 0.328·16-s − 1.12·17-s + 1.06·18-s − 0.676·19-s + 0.872·21-s − 0.901·22-s + 2.30·23-s + 1.68·24-s + 0.882·26-s − 0.769·27-s + 0.472·28-s − 0.397·29-s − 0.782·31-s + 0.480·32-s + 0.981·33-s − 1.19·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.425473876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.425473876\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $C_4$ | \( 1 - 3 T + 7 p T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 31 T + 2796 T^{2} + 31 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 p T + 3546 T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 79 T + 8288 T^{2} + 79 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 56 T + 1174 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 254 T + 38015 T^{2} - 254 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 62 T + 19751 T^{2} + 62 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 135 T + 63420 T^{2} + 135 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 113 T + 102964 T^{2} - 113 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 235 T + 143042 T^{2} - 235 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 804 T + 306321 T^{2} - 804 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 152 T + 180510 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 149 T + 269640 T^{2} - 149 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 441 T + 273734 T^{2} + 441 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 223 T + 149646 T^{2} + 223 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1157 T + 871890 T^{2} - 1157 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 619 T + 576906 T^{2} - 619 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 268 T + 763078 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 427 T + 986572 T^{2} + 427 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1211 T + 720720 T^{2} - 1211 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 466 T + 306170 T^{2} - 466 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 172 T - 273626 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.85919753065442622498907170085, −10.59333444576037829179880456064, −9.601977875832480089579576227904, −9.468960479219499458394271749213, −8.849056421123131864142375321339, −8.778351498589352675205054411990, −7.81572400732461855890984495501, −7.39551954694108854298745981874, −6.77388928272800678244653833131, −6.40980288269180802578694535687, −5.72902192319760416974702302077, −5.67587338633174292972238141469, −4.85052483572484563262538781795, −4.75123181023720967039259971988, −3.93060693162802408963263995972, −3.77827571140795938008556959803, −2.86327279723027258438942442193, −2.25968941286165932325369206806, −0.929427952112897476085075556694, −0.43840591368660462215669846199,
0.43840591368660462215669846199, 0.929427952112897476085075556694, 2.25968941286165932325369206806, 2.86327279723027258438942442193, 3.77827571140795938008556959803, 3.93060693162802408963263995972, 4.75123181023720967039259971988, 4.85052483572484563262538781795, 5.67587338633174292972238141469, 5.72902192319760416974702302077, 6.40980288269180802578694535687, 6.77388928272800678244653833131, 7.39551954694108854298745981874, 7.81572400732461855890984495501, 8.778351498589352675205054411990, 8.849056421123131864142375321339, 9.468960479219499458394271749213, 9.601977875832480089579576227904, 10.59333444576037829179880456064, 10.85919753065442622498907170085
