| L(s) = 1 | + 4·5-s − 12·7-s − 2·11-s + 8·13-s − 16·17-s + 12·19-s − 12·23-s + 11·25-s + 8·31-s − 48·35-s − 24·37-s + 18·41-s − 14·43-s + 8·47-s + 71·49-s − 16·53-s − 8·55-s − 6·59-s − 12·61-s + 32·65-s + 2·67-s + 24·77-s − 22·83-s − 64·85-s − 96·91-s + 48·95-s − 2·97-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 4.53·7-s − 0.603·11-s + 2.21·13-s − 3.88·17-s + 2.75·19-s − 2.50·23-s + 11/5·25-s + 1.43·31-s − 8.11·35-s − 3.94·37-s + 2.81·41-s − 2.13·43-s + 1.16·47-s + 71/7·49-s − 2.19·53-s − 1.07·55-s − 0.781·59-s − 1.53·61-s + 3.96·65-s + 0.244·67-s + 2.73·77-s − 2.41·83-s − 6.94·85-s − 10.0·91-s + 4.92·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4740836890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4740836890\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 4 T + p T^{2} + 8 T^{3} - 44 T^{4} + 8 p T^{5} + p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 12 T + 73 T^{2} + 300 T^{3} + 912 T^{4} + 300 p T^{5} + 73 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 26 T^{3} - 32 T^{4} + 26 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 97 T^{2} + 588 T^{3} + 2976 T^{4} + 588 p T^{5} + 97 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 9 T^{2} + 156 T^{3} - 484 T^{4} + 156 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 8 T + 13 T^{2} + 88 T^{3} - 344 T^{4} + 88 p T^{5} + 13 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 2472 T^{3} + 16862 T^{4} + 2472 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 18 T + 181 T^{2} - 1314 T^{3} + 8076 T^{4} - 1314 p T^{5} + 181 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 14 T + 65 T^{2} - 474 T^{3} - 6280 T^{4} - 474 p T^{5} + 65 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 8 T - 19 T^{2} + 88 T^{3} + 1672 T^{4} + 88 p T^{5} - 19 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1264 T^{3} + 11806 T^{4} + 1264 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 6 T + 45 T^{2} + 462 T^{3} + 848 T^{4} + 462 p T^{5} + 45 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 261 T^{2} + 2088 T^{3} + 24452 T^{4} + 2088 p T^{5} + 261 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T + 65 T^{2} - 762 T^{3} + 2600 T^{4} - 762 p T^{5} + 65 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 23814 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 - 155 T^{2} + 17784 T^{4} - 155 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 22 T + 185 T^{2} - 290 T^{3} - 14024 T^{4} - 290 p T^{5} + 185 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 13094 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.45511842571102296900110495765, −6.37982838125132915661697156251, −6.29852304518973334001398976989, −5.97293396243320623592485619866, −5.87848156989204378673312811230, −5.86331573223766769079580336707, −5.65459380116066627054827743565, −5.10740455173611621331015768599, −5.00802930288598194060794768433, −4.60486895372323910836754502598, −4.55308548518387847710116764308, −3.98038449687516917327486690671, −3.94273074779008143021655469915, −3.60122844268871607785424687077, −3.43460322865860853170710630332, −3.25326565838557088583836862069, −2.82016777546869339458220504407, −2.77412209660101194973492465350, −2.74070165851088063151620858803, −2.12920109650909918470528262684, −1.93922166922713445916469938068, −1.50295639127559131735070041286, −1.32650019315594894267096988009, −0.49899007974051927870410641547, −0.19247719208642159013192524329,
0.19247719208642159013192524329, 0.49899007974051927870410641547, 1.32650019315594894267096988009, 1.50295639127559131735070041286, 1.93922166922713445916469938068, 2.12920109650909918470528262684, 2.74070165851088063151620858803, 2.77412209660101194973492465350, 2.82016777546869339458220504407, 3.25326565838557088583836862069, 3.43460322865860853170710630332, 3.60122844268871607785424687077, 3.94273074779008143021655469915, 3.98038449687516917327486690671, 4.55308548518387847710116764308, 4.60486895372323910836754502598, 5.00802930288598194060794768433, 5.10740455173611621331015768599, 5.65459380116066627054827743565, 5.86331573223766769079580336707, 5.87848156989204378673312811230, 5.97293396243320623592485619866, 6.29852304518973334001398976989, 6.37982838125132915661697156251, 6.45511842571102296900110495765
