| L(s) = 1 | + 64·2-s + 3.07e3·4-s + 1.31e5·8-s + 8.54e4·9-s − 1.48e6·13-s + 5.24e6·16-s − 2.83e6·17-s + 5.46e6·18-s + 1.95e7·25-s − 9.50e7·26-s + 2.01e8·32-s − 1.81e8·34-s + 2.62e8·36-s − 2.28e8·49-s + 1.25e9·50-s − 4.56e9·52-s + 1.61e9·53-s + 7.51e9·64-s − 8.72e9·68-s + 1.11e10·72-s + 3.81e9·81-s + 1.70e10·89-s − 1.46e10·98-s + 6.00e10·100-s − 1.78e10·101-s − 1.94e11·104-s + 1.03e11·106-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 1.44·9-s − 3.99·13-s + 5·16-s − 2·17-s + 2.89·18-s + 2·25-s − 7.99·26-s + 6·32-s − 4·34-s + 4.33·36-s − 0.809·49-s + 4·50-s − 11.9·52-s + 3.86·53-s + 7·64-s − 6·68-s + 5.78·72-s + 1.09·81-s + 3.05·89-s − 1.61·98-s + 6·100-s − 1.69·101-s − 15.9·104-s + 7.72·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(11.65921653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.65921653\) |
| \(L(6)\) |
|
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 | $C_1$ | \( ( 1 - p^{5} T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + p^{5} T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 85424 T^{2} + p^{20} T^{4} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 228560576 T^{2} + p^{20} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 46503187952 T^{2} + p^{20} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 742568 T + p^{10} T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6670174766048 T^{2} + p^{20} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1015329483064576 T^{2} + p^{20} T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 807749318 T + p^{10} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 6371231079121526176 T^{2} + p^{20} T^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 16630472056735761376 T^{2} + p^{20} T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 8542932352 T + p^{10} T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
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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
−12.96137674653485028989140281485, −12.57635832128367608277489824065, −11.92164799106934324505704426259, −11.81990826961108852546419966394, −10.71147455232044387013333683304, −10.38647009825004170525359642853, −9.870560953871413905444516873321, −9.081309636693300811235377581555, −7.83380632169992524342339998486, −7.11523251700564506902916567272, −7.06657946739300630313298230192, −6.56651524027178804588333518744, −5.27420941574072731511662012271, −4.89231307721040062376531959055, −4.54846684572329702367994008906, −3.94410924530090777971462710388, −2.63774117773464714304214533446, −2.49826111539994365209531199977, −1.81898169544823792242296916451, −0.62429920822363523484301381551,
0.62429920822363523484301381551, 1.81898169544823792242296916451, 2.49826111539994365209531199977, 2.63774117773464714304214533446, 3.94410924530090777971462710388, 4.54846684572329702367994008906, 4.89231307721040062376531959055, 5.27420941574072731511662012271, 6.56651524027178804588333518744, 7.06657946739300630313298230192, 7.11523251700564506902916567272, 7.83380632169992524342339998486, 9.081309636693300811235377581555, 9.870560953871413905444516873321, 10.38647009825004170525359642853, 10.71147455232044387013333683304, 11.81990826961108852546419966394, 11.92164799106934324505704426259, 12.57635832128367608277489824065, 12.96137674653485028989140281485
