| L(s) = 1 | + 2·2-s − 3·3-s + 6·5-s − 6·6-s − 7·7-s − 8·8-s + 12·10-s + 30·11-s + 106·13-s − 14·14-s − 18·15-s − 16·16-s + 84·17-s + 97·19-s + 21·21-s + 60·22-s − 84·23-s + 24·24-s + 125·25-s + 212·26-s + 27·27-s − 360·29-s − 36·30-s − 179·31-s − 90·33-s + 168·34-s − 42·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.536·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.379·10-s + 0.822·11-s + 2.26·13-s − 0.267·14-s − 0.309·15-s − 1/4·16-s + 1.19·17-s + 1.17·19-s + 0.218·21-s + 0.581·22-s − 0.761·23-s + 0.204·24-s + 25-s + 1.59·26-s + 0.192·27-s − 2.30·29-s − 0.219·30-s − 1.03·31-s − 0.474·33-s + 0.847·34-s − 0.202·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.015328069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.015328069\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 + p T + p^{3} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 6 T - 89 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 30 T - 431 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 53 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 84 T + 2143 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 97 T + 2550 T^{2} - 97 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 84 T - 5111 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 180 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 179 T + 2250 T^{2} + 179 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 145 T - 29628 T^{2} - 145 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 126 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 325 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 366 T + 30133 T^{2} - 366 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 768 T + 440947 T^{2} - 768 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 264 T - 135683 T^{2} - 264 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 818 T + 442143 T^{2} + 818 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 523 T - 27234 T^{2} - 523 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 342 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 43 T - 387168 T^{2} - 43 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 1171 T + 878202 T^{2} - 1171 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 810 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 600 T - 344969 T^{2} - 600 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 386 T + p^{3} T^{2} )^{2} \) |
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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
−16.01504615626369712381374530755, −14.97491480999937009192244553923, −14.73023318363404129964359223639, −13.87588602038084966240040601458, −13.52866516726827855239645999132, −13.02921865609373329329579525012, −12.33873985931540435304269233141, −11.67643968516873195857533764868, −11.19343967457713905533686560717, −10.54429528617865605778326017431, −9.626733058347137120525506564114, −9.183894000260580074093694247115, −8.373626041104596179888390215198, −7.35047050678237461859203088133, −6.44012119996466410374572030571, −5.74247076804874984899918938320, −5.45277309721946286272627342430, −3.94025210454468373485611824610, −3.39307601362238506410893754220, −1.32989402737123080357621552300,
1.32989402737123080357621552300, 3.39307601362238506410893754220, 3.94025210454468373485611824610, 5.45277309721946286272627342430, 5.74247076804874984899918938320, 6.44012119996466410374572030571, 7.35047050678237461859203088133, 8.373626041104596179888390215198, 9.183894000260580074093694247115, 9.626733058347137120525506564114, 10.54429528617865605778326017431, 11.19343967457713905533686560717, 11.67643968516873195857533764868, 12.33873985931540435304269233141, 13.02921865609373329329579525012, 13.52866516726827855239645999132, 13.87588602038084966240040601458, 14.73023318363404129964359223639, 14.97491480999937009192244553923, 16.01504615626369712381374530755
