| L(s) = 1 | − 4·2-s + 12·4-s − 12·5-s − 32·8-s + 48·10-s − 4·11-s + 48·13-s + 80·16-s − 132·17-s + 120·19-s − 144·20-s + 16·22-s + 76·23-s − 140·25-s − 192·26-s + 112·29-s + 432·31-s − 192·32-s + 528·34-s − 280·37-s − 480·38-s + 384·40-s − 36·41-s − 128·43-s − 48·44-s − 304·46-s + 264·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.07·5-s − 1.41·8-s + 1.51·10-s − 0.109·11-s + 1.02·13-s + 5/4·16-s − 1.88·17-s + 1.44·19-s − 1.60·20-s + 0.155·22-s + 0.689·23-s − 1.11·25-s − 1.44·26-s + 0.717·29-s + 2.50·31-s − 1.06·32-s + 2.66·34-s − 1.24·37-s − 2.04·38-s + 1.51·40-s − 0.137·41-s − 0.453·43-s − 0.164·44-s − 0.974·46-s + 0.819·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 12 T + 284 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 2594 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 48 T + 4520 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 132 T + 820 p T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 120 T + 13086 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 76 T + 4970 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 112 T + 6914 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 432 T + 100406 T^{2} - 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 280 T + 56106 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 36 T + 105908 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 128 T - 2778 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 264 T - 3418 T^{2} - 264 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 268 T + 241982 T^{2} + 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 336 T + 261374 T^{2} + 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 504 T + 268248 T^{2} + 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 384 T + 596918 T^{2} + 384 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 396 T + 521098 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 312 T + 94320 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 848 T + 985854 T^{2} + 848 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 648 T + 1235750 T^{2} - 648 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 612 T + 1442324 T^{2} - 612 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2184 T + 2982432 T^{2} + 2184 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.329378591428063918216598753755, −9.155660751676870838734898390465, −8.649401809578329842333574905679, −8.241555674483904794557519650669, −8.078007338646603122008562390389, −7.43968822219664070941235706518, −7.23561181822408275497593318828, −6.59907955931469152228777125575, −6.28641875749041656283365060356, −5.90801905790919284798576446434, −4.92393863039880438068452496732, −4.77515703539850638916203129980, −3.80484858623199286895703159076, −3.68945485053633272072374993435, −2.68632041969242410553035852191, −2.58551049044513897269674910585, −1.31475468555024002577038910468, −1.25251076799397200600005695343, 0, 0,
1.25251076799397200600005695343, 1.31475468555024002577038910468, 2.58551049044513897269674910585, 2.68632041969242410553035852191, 3.68945485053633272072374993435, 3.80484858623199286895703159076, 4.77515703539850638916203129980, 4.92393863039880438068452496732, 5.90801905790919284798576446434, 6.28641875749041656283365060356, 6.59907955931469152228777125575, 7.23561181822408275497593318828, 7.43968822219664070941235706518, 8.078007338646603122008562390389, 8.241555674483904794557519650669, 8.649401809578329842333574905679, 9.155660751676870838734898390465, 9.329378591428063918216598753755
