| L(s) = 1 | + 2·2-s − 2·3-s + 4-s + 2·5-s − 4·6-s − 7-s − 2·8-s + 9-s + 4·10-s + 6·11-s − 2·12-s + 2·13-s − 2·14-s − 4·15-s − 4·16-s + 12·17-s + 2·18-s − 7·19-s + 2·20-s + 2·21-s + 12·22-s + 18·23-s + 4·24-s + 25-s + 4·26-s + 2·27-s − 28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s − 1.63·6-s − 0.377·7-s − 0.707·8-s + 1/3·9-s + 1.26·10-s + 1.80·11-s − 0.577·12-s + 0.554·13-s − 0.534·14-s − 1.03·15-s − 16-s + 2.91·17-s + 0.471·18-s − 1.60·19-s + 0.447·20-s + 0.436·21-s + 2.55·22-s + 3.75·23-s + 0.816·24-s + 1/5·25-s + 0.784·26-s + 0.384·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.763625681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.763625681\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + T - 5 T^{2} - 8 T^{3} - 20 T^{4} - 8 p T^{5} - 5 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 7 T + 7 T^{2} + 28 T^{3} + 472 T^{4} + 28 p T^{5} + 7 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 6 T - 22 T^{2} - 144 T^{3} + 207 T^{4} - 144 p T^{5} - 22 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 9 T + 106 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 6 T - 46 T^{2} - 144 T^{3} + 2007 T^{4} - 144 p T^{5} - 46 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 3 T - 103 T^{2} + 18 T^{3} + 8532 T^{4} + 18 p T^{5} - 103 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 13 T + T^{2} + 442 T^{3} + 12412 T^{4} + 442 p T^{5} + p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 6 T - 82 T^{2} - 144 T^{3} + 6327 T^{4} - 144 p T^{5} - 82 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 5 T + 78 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 17 T + 133 T^{2} + 34 T^{3} - 4736 T^{4} + 34 p T^{5} + 133 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^3$ | \( 1 - 34 T^{2} - 5733 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 15 T + 65 T^{2} + 270 T^{3} - 2346 T^{4} + 270 p T^{5} + 65 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 19 T + 210 T^{2} - 19 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.96726428447407423518085692492, −6.52979496259404026868994716283, −6.46769858193064993486654250708, −6.24635682632801241006633661096, −6.17545599783337158385751155978, −5.86068786626288978992305006018, −5.65927790242469358274491317523, −5.44201385356941261467740275265, −5.29276472123007053112734641090, −5.08094175018132647945746660643, −4.80616733257135348706992170716, −4.70285079084255882121257682642, −4.12315414765356406727162364589, −4.02151685281758240390219948833, −3.97276363144708872972146069774, −3.50525203871378483555146796037, −3.20238884400745860649102043135, −3.06807475985732487721816189469, −2.98690837313972318832557579844, −2.44037291519821082605602516359, −1.84464081723413483553197368337, −1.79398152111434136095456890858, −1.09712061038837054007181765020, −1.06007863371185985307460205383, −0.55822790667070444982146460398,
0.55822790667070444982146460398, 1.06007863371185985307460205383, 1.09712061038837054007181765020, 1.79398152111434136095456890858, 1.84464081723413483553197368337, 2.44037291519821082605602516359, 2.98690837313972318832557579844, 3.06807475985732487721816189469, 3.20238884400745860649102043135, 3.50525203871378483555146796037, 3.97276363144708872972146069774, 4.02151685281758240390219948833, 4.12315414765356406727162364589, 4.70285079084255882121257682642, 4.80616733257135348706992170716, 5.08094175018132647945746660643, 5.29276472123007053112734641090, 5.44201385356941261467740275265, 5.65927790242469358274491317523, 5.86068786626288978992305006018, 6.17545599783337158385751155978, 6.24635682632801241006633661096, 6.46769858193064993486654250708, 6.52979496259404026868994716283, 6.96726428447407423518085692492
