| L(s) = 1 | + 520·4-s + 1.52e3·7-s + 1.46e5·13-s + 2.62e5·16-s − 2.39e6·19-s + 3.66e6·25-s + 7.93e5·28-s + 3.64e6·31-s + 5.70e7·37-s − 1.55e7·43-s + 8.12e7·49-s + 7.59e7·52-s + 3.05e8·61-s + 2.68e8·64-s + 6.41e8·67-s + 2.65e8·73-s − 1.24e9·76-s + 2.30e8·79-s + 2.22e8·91-s + 4.70e8·97-s + 1.90e9·100-s − 2.01e8·103-s − 9.06e9·109-s + 4.00e8·112-s + 1.47e9·121-s + 1.89e9·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1.01·4-s + 0.240·7-s + 1.41·13-s + 16-s − 4.21·19-s + 1.87·25-s + 0.243·28-s + 0.709·31-s + 5.00·37-s − 0.692·43-s + 2.01·49-s + 1.44·52-s + 2.82·61-s + 1.99·64-s + 3.88·67-s + 1.09·73-s − 4.27·76-s + 0.666·79-s + 0.340·91-s + 0.539·97-s + 1.90·100-s − 0.176·103-s − 6.15·109-s + 0.240·112-s + 0.627·121-s + 0.720·124-s − 1.01·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(12.83776827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.83776827\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 65 p^{3} T^{2} + 129 p^{6} T^{4} - 65 p^{21} T^{6} + p^{36} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - 3662314 T^{2} + 9597846568971 T^{4} - 3662314 p^{18} T^{6} + p^{36} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 109 p T - 811662 p^{2} T^{2} - 109 p^{10} T^{3} + p^{18} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 1479632758 T^{2} - 3370604214945544917 T^{4} - 1479632758 p^{18} T^{6} + p^{36} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 73015 T - 5273309148 T^{2} - 73015 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 208871094850 T^{2} + p^{18} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 598129 T + p^{9} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 2159580975410 T^{2} + \)\(14\!\cdots\!31\)\( T^{4} + 2159580975410 p^{18} T^{6} + p^{36} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 7356589617274 T^{2} - \)\(15\!\cdots\!85\)\( T^{4} - 7356589617274 p^{18} T^{6} + p^{36} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 1824100 T - 23112281350671 T^{2} - 1824100 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 14272175 T + p^{9} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 234725122853678 T^{2} - \)\(52\!\cdots\!37\)\( T^{4} + 234725122853678 p^{18} T^{6} + p^{36} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 7756904 T - 442423052271627 T^{2} + 7756904 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 1277861803424638 T^{2} + \)\(38\!\cdots\!55\)\( T^{4} - 1277861803424638 p^{18} T^{6} + p^{36} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 6501492755943082 T^{2} + p^{18} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 1648507597908694 T^{2} - \)\(72\!\cdots\!85\)\( T^{4} - 1648507597908694 p^{18} T^{6} + p^{36} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 152766493 T + 11643455290684908 T^{2} - 152766493 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 320752069 T + 75675355371485814 T^{2} - 320752069 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 79782903657305998 T^{2} + p^{18} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 66463985 T + p^{9} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 115357453 T - 106544254019971110 T^{2} - 115357453 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 3900339259244570 T^{2} - \)\(34\!\cdots\!09\)\( T^{4} + 3900339259244570 p^{18} T^{6} + p^{36} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 586605998589983314 T^{2} + p^{18} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 235306843 T - 704861748291938568 T^{2} - 235306843 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
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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
−8.745738626880019683735560242230, −8.216317923931249081139654581912, −8.133863032976531301380379137263, −7.994990777228925649329172152978, −7.66606129347083113351472429584, −6.84152507584236518576777093750, −6.63547446718084765628982677954, −6.60339080968757616424987224872, −6.44576941109770765437618442481, −6.08706925241758656838181449289, −5.55615943962965122444971382350, −5.28381883090985909934251568145, −4.94483284613005454803747105552, −4.21002221682861906350523395200, −4.08409916440918738605021116212, −4.01865939153061404403754215523, −3.57630027374152330617356071093, −2.56226523617330613506608653449, −2.54930247512194649086234564507, −2.53915337452782874455703772061, −2.04184661868414367835218443316, −1.37742020021890269004177883361, −1.05841402512090494593719708471, −0.61110953830559006317461109041, −0.55760579713395727671144166622,
0.55760579713395727671144166622, 0.61110953830559006317461109041, 1.05841402512090494593719708471, 1.37742020021890269004177883361, 2.04184661868414367835218443316, 2.53915337452782874455703772061, 2.54930247512194649086234564507, 2.56226523617330613506608653449, 3.57630027374152330617356071093, 4.01865939153061404403754215523, 4.08409916440918738605021116212, 4.21002221682861906350523395200, 4.94483284613005454803747105552, 5.28381883090985909934251568145, 5.55615943962965122444971382350, 6.08706925241758656838181449289, 6.44576941109770765437618442481, 6.60339080968757616424987224872, 6.63547446718084765628982677954, 6.84152507584236518576777093750, 7.66606129347083113351472429584, 7.994990777228925649329172152978, 8.133863032976531301380379137263, 8.216317923931249081139654581912, 8.745738626880019683735560242230
