| L(s) = 1 | + 128·4-s + 572·7-s + 1.22e4·16-s + 1.05e4·19-s − 3.12e4·25-s + 7.32e4·28-s + 2.22e5·43-s + 1.00e4·49-s + 8.41e5·61-s + 1.04e6·64-s + 1.27e6·73-s + 1.35e6·76-s − 4.00e6·100-s + 7.02e6·112-s − 3.54e6·121-s + 127-s + 131-s + 6.05e6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 9.39e6·169-s + 2.85e7·172-s + ⋯ |
| L(s) = 1 | + 2·4-s + 1.66·7-s + 3·16-s + 1.54·19-s − 2·25-s + 3.33·28-s + 2.80·43-s + 0.0857·49-s + 3.70·61-s + 4·64-s + 3.28·73-s + 3.08·76-s − 4·100-s + 5.00·112-s − 2·121-s + 2.57·133-s − 1.94·169-s + 5.60·172-s − 3.33·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(8.671525440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.671525440\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 10582 T + p^{6} T^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 286 T + p^{6} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 506 T + p^{6} T^{2} )( 1 + 506 T + p^{6} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 35282 T + p^{6} T^{2} )( 1 + 35282 T + p^{6} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 89206 T + p^{6} T^{2} )( 1 + 89206 T + p^{6} T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 111386 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 420838 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 172874 T + p^{6} T^{2} )( 1 + 172874 T + p^{6} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 638066 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 204622 T + p^{6} T^{2} )( 1 + 204622 T + p^{6} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 56446 T + p^{6} T^{2} )( 1 + 56446 T + p^{6} T^{2} ) \) |
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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
−11.51762495810427438238329927287, −11.49644638352365269778620032988, −11.14108213430189616092721372645, −10.66348473604183985594044338417, −9.791625138177621501965715835725, −9.762943236538033874907092427648, −8.672333265011621761747766825061, −7.968856438469336108193369901397, −7.72773497901759778851707127256, −7.42403834433641781263149321181, −6.68218196134577871547036845040, −6.12491062249722595874107162563, −5.34870774228365096501112635908, −5.26072419504278525002148555225, −4.00519243481250847016500072127, −3.51871650779186558592919661517, −2.37041865056747335552927195358, −2.25794722430316225200980984497, −1.36338936140429378843577212029, −0.889893865221661034807282066488,
0.889893865221661034807282066488, 1.36338936140429378843577212029, 2.25794722430316225200980984497, 2.37041865056747335552927195358, 3.51871650779186558592919661517, 4.00519243481250847016500072127, 5.26072419504278525002148555225, 5.34870774228365096501112635908, 6.12491062249722595874107162563, 6.68218196134577871547036845040, 7.42403834433641781263149321181, 7.72773497901759778851707127256, 7.968856438469336108193369901397, 8.672333265011621761747766825061, 9.762943236538033874907092427648, 9.791625138177621501965715835725, 10.66348473604183985594044338417, 11.14108213430189616092721372645, 11.49644638352365269778620032988, 11.51762495810427438238329927287
