| L(s) = 1 | − 16·2-s − 61·3-s + 192·4-s + 175·5-s + 976·6-s − 2.59e3·7-s − 2.04e3·8-s − 899·9-s − 2.80e3·10-s + 1.04e3·11-s − 1.17e4·12-s + 1.46e4·13-s + 4.14e4·14-s − 1.06e4·15-s + 2.04e4·16-s + 2.96e4·17-s + 1.43e4·18-s − 1.37e4·19-s + 3.36e4·20-s + 1.58e5·21-s − 1.67e4·22-s + 6.99e4·23-s + 1.24e5·24-s + 2.06e4·25-s − 2.34e5·26-s + 2.03e5·27-s − 4.97e5·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.30·3-s + 3/2·4-s + 0.626·5-s + 1.84·6-s − 2.85·7-s − 1.41·8-s − 0.411·9-s − 0.885·10-s + 0.236·11-s − 1.95·12-s + 1.84·13-s + 4.03·14-s − 0.816·15-s + 5/4·16-s + 1.46·17-s + 0.581·18-s − 0.458·19-s + 0.939·20-s + 3.72·21-s − 0.334·22-s + 1.19·23-s + 1.84·24-s + 0.264·25-s − 2.61·26-s + 1.98·27-s − 4.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.2959885941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2959885941\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 61 T + 1540 p T^{2} + 61 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 7 p^{2} T + 398 p^{2} T^{2} - 7 p^{9} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2592 T + 3302069 T^{2} + 2592 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 95 p T - 11749120 T^{2} - 95 p^{8} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 14647 T + 173951598 T^{2} - 14647 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 29616 T + 1034411785 T^{2} - 29616 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 69985 T + 6725368502 T^{2} - 69985 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 138821 T + 15732402524 T^{2} - 138821 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 199396 T + 48643192158 T^{2} + 199396 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 67840 T + 47468074998 T^{2} + 67840 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 539350 T + 389443599074 T^{2} + 539350 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 602639 T + 634135307466 T^{2} - 602639 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1031975 T + 749722035926 T^{2} + 1031975 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2138263 T + 2496539389382 T^{2} - 2138263 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3936369 T + 8841477061504 T^{2} + 3936369 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1027655 T + 2220629234304 T^{2} + 1027655 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 764949 T + 9626156199098 T^{2} - 764949 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3572084 T + 9205034831954 T^{2} + 3572084 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9069522 T + 42655939394627 T^{2} - 9069522 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2753414 T - 12495532808130 T^{2} + 2753414 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 7643046 T + 68606421453310 T^{2} + 7643046 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1393620 T + 87948683189458 T^{2} - 1393620 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6921466 T + 112774027874802 T^{2} + 6921466 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−15.60371112166313091764822822264, −14.64207489531025652892322361865, −13.67897111515672893487798562660, −13.14988329594812363786290642313, −12.34989896592906312005896818868, −12.12377651495366380373995357089, −11.00933838033197834409507315237, −10.79301479765061739442322805669, −10.08588130994242838233765638962, −9.506723074617059862670597511685, −8.979934108385818725973002491780, −8.331890805388817026525985464913, −6.88854766110946134361761071661, −6.57715966607988348260165599917, −5.91709819697111846823935045847, −5.64372488215454239126092490370, −3.38305678550658870432199098515, −2.99613492014459687651056813198, −1.17816343943625185396063877845, −0.37490776553246150925276510491,
0.37490776553246150925276510491, 1.17816343943625185396063877845, 2.99613492014459687651056813198, 3.38305678550658870432199098515, 5.64372488215454239126092490370, 5.91709819697111846823935045847, 6.57715966607988348260165599917, 6.88854766110946134361761071661, 8.331890805388817026525985464913, 8.979934108385818725973002491780, 9.506723074617059862670597511685, 10.08588130994242838233765638962, 10.79301479765061739442322805669, 11.00933838033197834409507315237, 12.12377651495366380373995357089, 12.34989896592906312005896818868, 13.14988329594812363786290642313, 13.67897111515672893487798562660, 14.64207489531025652892322361865, 15.60371112166313091764822822264
