| L(s) = 1 | − 4·2-s − 6·3-s + 12·4-s + 24·6-s + 43·7-s − 32·8-s + 27·9-s − 66·11-s − 72·12-s + 16·13-s − 172·14-s + 80·16-s − 27·17-s − 108·18-s − 40·19-s − 258·21-s + 264·22-s + 16·23-s + 192·24-s − 64·26-s − 108·27-s + 516·28-s − 440·29-s − 201·31-s − 192·32-s + 396·33-s + 108·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s + 2.32·7-s − 1.41·8-s + 9-s − 1.80·11-s − 1.73·12-s + 0.341·13-s − 3.28·14-s + 5/4·16-s − 0.385·17-s − 1.41·18-s − 0.482·19-s − 2.68·21-s + 2.55·22-s + 0.145·23-s + 1.63·24-s − 0.482·26-s − 0.769·27-s + 3.48·28-s − 2.81·29-s − 1.16·31-s − 1.06·32-s + 2.08·33-s + 0.544·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 562500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 562500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 - 43 T + 1047 T^{2} - 43 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 p T + 3671 T^{2} + 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 16 T + 2653 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 27 T + 9347 T^{2} + 27 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 40 T + 13873 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 16 T + 20193 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 440 T + 96333 T^{2} + 440 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 201 T + 50771 T^{2} + 201 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 428 T + 146857 T^{2} - 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 171 T + 3151 p T^{2} + 171 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 431 T + 108153 T^{2} - 431 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 388 T + 151437 T^{2} - 388 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 281 T - 21807 T^{2} - 281 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 205 T + 162483 T^{2} + 205 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 489 T + 354431 T^{2} - 489 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 188 T + 593542 T^{2} - 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 771 T + 842321 T^{2} + 771 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1169 T + 963913 T^{2} + 1169 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 930 T + 975458 T^{2} + 930 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 39 T + 1085093 T^{2} + 39 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 310 T + 711963 T^{2} + 310 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2177 T + 2898427 T^{2} + 2177 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.565669767172451500426728395585, −9.549445173439176769044215200746, −8.630818101078618864262549654007, −8.615458462142754387065485773611, −7.899136081010759179527024920301, −7.64077316692004557583866552860, −7.36316051445170267557488630710, −7.02436046531270351930290647262, −6.00060324384334997777608813100, −5.83962061444360747939641524152, −5.19563257949940631237451146353, −5.17712914589127180722724385033, −4.14827676589250287708020817051, −4.01846241416800244555362731934, −2.53505403751671984675172604389, −2.39181356855657784603154106496, −1.42484269020431101846874105843, −1.32057028105674229243694822642, 0, 0,
1.32057028105674229243694822642, 1.42484269020431101846874105843, 2.39181356855657784603154106496, 2.53505403751671984675172604389, 4.01846241416800244555362731934, 4.14827676589250287708020817051, 5.17712914589127180722724385033, 5.19563257949940631237451146353, 5.83962061444360747939641524152, 6.00060324384334997777608813100, 7.02436046531270351930290647262, 7.36316051445170267557488630710, 7.64077316692004557583866552860, 7.899136081010759179527024920301, 8.615458462142754387065485773611, 8.630818101078618864262549654007, 9.549445173439176769044215200746, 9.565669767172451500426728395585
