| L(s) = 1 | − 488·7-s + 5.45e3·13-s + 1.07e4·19-s + 2.72e4·25-s + 2.03e4·31-s − 1.30e5·37-s + 9.89e4·43-s − 5.66e4·49-s − 2.01e5·61-s + 8.71e5·67-s + 1.23e6·73-s + 1.02e6·79-s − 2.66e6·91-s + 8.54e4·97-s − 2.84e6·103-s − 2.80e5·109-s − 2.18e6·121-s + 127-s + 131-s − 5.26e6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.42·7-s + 2.48·13-s + 1.57·19-s + 1.74·25-s + 0.682·31-s − 2.56·37-s + 1.24·43-s − 0.481·49-s − 0.886·61-s + 2.89·67-s + 3.18·73-s + 2.08·79-s − 3.53·91-s + 0.0935·97-s − 2.60·103-s − 0.216·109-s − 1.23·121-s − 2.23·133-s + 2.62·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(4.540932023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.540932023\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 1088 p^{2} T^{2} + p^{12} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 244 T + p^{6} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2182606 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2728 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 13547360 T^{2} + p^{12} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5392 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 289848386 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 575602720 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10172 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 65006 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 4637059040 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 49480 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 5600983390 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 43709554208 T^{2} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 44614042990 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 100610 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 435736 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 252585347330 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 619568 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 514340 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 594384805586 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 810559043264 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 42704 T + p^{6} T^{2} )^{2} \) |
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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
−9.828762540520643926766128979614, −9.457105798709890241754119033513, −9.273839071024278036994059086592, −8.677713970146571293855110860460, −8.175867647277724054696468937001, −8.034390956351621362796610108204, −7.04340797846624007999338911501, −6.74158748813441660181753454624, −6.53708227527787902233998684828, −5.99721671789249366387240888699, −5.30256569133217094156740353922, −5.19055610461383027106015652110, −4.24455307314706994255815944889, −3.64697490681104820120824959430, −3.23064984707884023117714258736, −3.16911638035349727222953282521, −2.17866364484372133077863214196, −1.44503135750968085865721221649, −0.807685145094876948717986234945, −0.59874340492798756618860875400,
0.59874340492798756618860875400, 0.807685145094876948717986234945, 1.44503135750968085865721221649, 2.17866364484372133077863214196, 3.16911638035349727222953282521, 3.23064984707884023117714258736, 3.64697490681104820120824959430, 4.24455307314706994255815944889, 5.19055610461383027106015652110, 5.30256569133217094156740353922, 5.99721671789249366387240888699, 6.53708227527787902233998684828, 6.74158748813441660181753454624, 7.04340797846624007999338911501, 8.034390956351621362796610108204, 8.175867647277724054696468937001, 8.677713970146571293855110860460, 9.273839071024278036994059086592, 9.457105798709890241754119033513, 9.828762540520643926766128979614
