| L(s) = 1 | − 4·2-s − 48·4-s + 448·8-s + 498·9-s − 2.93e3·13-s + 1.28e3·16-s + 9.53e3·17-s − 1.99e3·18-s + 1.17e4·26-s + 5.09e4·29-s − 3.37e4·32-s − 3.81e4·34-s − 2.39e4·36-s − 3.98e3·37-s + 5.87e4·41-s + 1.39e5·49-s + 1.40e5·52-s + 3.85e5·53-s − 2.03e5·58-s − 2.18e4·61-s + 5.32e4·64-s − 4.57e5·68-s + 2.23e5·72-s − 5.77e5·73-s + 1.59e4·74-s − 2.83e5·81-s − 2.34e5·82-s + ⋯ |
| L(s) = 1 | − 1/2·2-s − 3/4·4-s + 7/8·8-s + 0.683·9-s − 1.33·13-s + 5/16·16-s + 1.94·17-s − 0.341·18-s + 0.667·26-s + 2.09·29-s − 1.03·32-s − 0.970·34-s − 0.512·36-s − 0.0787·37-s + 0.852·41-s + 1.18·49-s + 1.00·52-s + 2.59·53-s − 1.04·58-s − 0.0962·61-s + 0.203·64-s − 1.45·68-s + 0.597·72-s − 1.48·73-s + 0.0393·74-s − 0.533·81-s − 0.426·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.922237431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.922237431\) |
| \(L(4)\) |
|
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 | $C_2$ | \( 1 + p^{2} T + p^{6} T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 166 p T^{2} + p^{12} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 139298 T^{2} + p^{12} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2620562 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 1466 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4766 T + p^{6} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 37404722 T^{2} + p^{12} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 186397538 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 25498 T + p^{6} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 20219522 T^{2} + p^{12} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 1994 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 29362 T + p^{6} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 12179022098 T^{2} + p^{12} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 21501276098 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 192854 T + p^{6} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 78211344722 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10918 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 25565876978 T^{2} + p^{12} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 27079768798 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 288626 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 389636558402 T^{2} + p^{12} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 612202422578 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 310738 T + p^{6} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1457086 T + p^{6} T^{2} )^{2} \) |
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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
−12.69538817065064461822306441658, −12.57199759004697438714482682106, −11.98435732301681267878677398851, −11.48611020701202545169041995744, −10.27198648258874423863541295121, −10.14678712266239335333854943480, −10.08126348172967847569334761619, −9.043725963544900981115941502752, −8.760171459806232266670932038422, −7.88101053798562897690262562458, −7.50274177189736997003943230820, −7.05407740591684057546484934350, −6.01468497622942802714446379397, −5.32437986372162815565070486840, −4.71480608547640693120748589612, −4.11281181211473746291016240871, −3.21266746013634129799978007470, −2.28791116608141135436355143594, −1.11735431089136834325813856640, −0.64676256766681732266851722257,
0.64676256766681732266851722257, 1.11735431089136834325813856640, 2.28791116608141135436355143594, 3.21266746013634129799978007470, 4.11281181211473746291016240871, 4.71480608547640693120748589612, 5.32437986372162815565070486840, 6.01468497622942802714446379397, 7.05407740591684057546484934350, 7.50274177189736997003943230820, 7.88101053798562897690262562458, 8.760171459806232266670932038422, 9.043725963544900981115941502752, 10.08126348172967847569334761619, 10.14678712266239335333854943480, 10.27198648258874423863541295121, 11.48611020701202545169041995744, 11.98435732301681267878677398851, 12.57199759004697438714482682106, 12.69538817065064461822306441658
