| L(s) = 1 | + 4·5-s − 4·7-s − 2·9-s − 8·11-s − 4·13-s − 8·19-s − 12·23-s + 10·25-s + 8·27-s + 4·29-s + 16·31-s − 16·35-s − 4·37-s − 12·41-s + 16·43-s − 8·45-s + 8·49-s − 4·53-s − 32·55-s − 16·59-s − 4·61-s + 8·63-s − 16·65-s − 8·67-s + 12·71-s + 28·73-s + 32·77-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.51·7-s − 2/3·9-s − 2.41·11-s − 1.10·13-s − 1.83·19-s − 2.50·23-s + 2·25-s + 1.53·27-s + 0.742·29-s + 2.87·31-s − 2.70·35-s − 0.657·37-s − 1.87·41-s + 2.43·43-s − 1.19·45-s + 8/7·49-s − 0.549·53-s − 4.31·55-s − 2.08·59-s − 0.512·61-s + 1.00·63-s − 1.98·65-s − 0.977·67-s + 1.42·71-s + 3.27·73-s + 3.64·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7096370182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7096370182\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} ) \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 - 8 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + 6 T + p T^{2} )^{2}( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $D_4\times C_2$ | \( 1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 4 p T^{5} + 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 8 T + 18 T^{2} - 160 T^{3} - 1246 T^{4} - 160 p T^{5} + 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 300 T^{3} + 1246 T^{4} + 300 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} + 204 T^{3} - 830 T^{4} + 204 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 4 p T^{5} + 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 516 T^{3} + 3694 T^{4} + 516 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 16 T + 162 T^{2} - 1384 T^{3} + 10178 T^{4} - 1384 p T^{5} + 162 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 486 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 4 T + 54 T^{2} + 708 T^{3} + 3490 T^{4} + 708 p T^{5} + 54 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 16 T + 114 T^{2} + 696 T^{3} + 4834 T^{4} + 696 p T^{5} + 114 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 4 p T^{5} + 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 8 T + 18 T^{2} - 736 T^{3} - 5854 T^{4} - 736 p T^{5} + 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 876 T^{3} + 10654 T^{4} - 876 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16 T + 114 T^{2} - 792 T^{3} + 6370 T^{4} - 792 p T^{5} + 114 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 516 T^{3} + 1582 T^{4} - 516 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.724129027364337090362334196950, −8.612489646639570121176308646373, −8.171849489025051343610788359363, −8.046690377839520544090238194354, −7.81784740065322504792171649381, −7.63025063119180227594362763050, −7.01486095300628755445851119949, −6.73647444510445565589765587313, −6.52519675176320441773628871847, −6.30909732504273133543872004360, −6.14019827252509520796946122644, −5.99248497263894253253794658931, −5.42310374656439398486846531162, −5.33984877482147379523385941382, −4.84931891560977910494127252159, −4.73713303565563725836505926758, −4.49884857309430869895800454415, −3.79262279164531121409164249491, −3.57789961753940801480259924595, −2.71823636730398433863631993334, −2.69938179409490238225491143709, −2.63129916045971228290448228239, −2.27388619249460429245900498453, −1.62456651384583248660789909657, −0.40080424651910957333917738776,
0.40080424651910957333917738776, 1.62456651384583248660789909657, 2.27388619249460429245900498453, 2.63129916045971228290448228239, 2.69938179409490238225491143709, 2.71823636730398433863631993334, 3.57789961753940801480259924595, 3.79262279164531121409164249491, 4.49884857309430869895800454415, 4.73713303565563725836505926758, 4.84931891560977910494127252159, 5.33984877482147379523385941382, 5.42310374656439398486846531162, 5.99248497263894253253794658931, 6.14019827252509520796946122644, 6.30909732504273133543872004360, 6.52519675176320441773628871847, 6.73647444510445565589765587313, 7.01486095300628755445851119949, 7.63025063119180227594362763050, 7.81784740065322504792171649381, 8.046690377839520544090238194354, 8.171849489025051343610788359363, 8.612489646639570121176308646373, 8.724129027364337090362334196950
