| L(s) = 1 | + 4·2-s + 8·4-s + 4·5-s − 4·7-s + 8·8-s + 2·9-s + 16·10-s + 4·11-s − 12·13-s − 16·14-s − 4·16-s + 8·18-s − 4·19-s + 32·20-s + 16·22-s + 8·25-s − 48·26-s − 32·28-s + 12·29-s − 32·32-s − 16·35-s + 16·36-s − 4·37-s − 16·38-s + 32·40-s + 24·41-s − 4·43-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4·4-s + 1.78·5-s − 1.51·7-s + 2.82·8-s + 2/3·9-s + 5.05·10-s + 1.20·11-s − 3.32·13-s − 4.27·14-s − 16-s + 1.88·18-s − 0.917·19-s + 7.15·20-s + 3.41·22-s + 8/5·25-s − 9.41·26-s − 6.04·28-s + 2.22·29-s − 5.65·32-s − 2.70·35-s + 8/3·36-s − 0.657·37-s − 2.59·38-s + 5.05·40-s + 3.74·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.993945847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.993945847\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 12 T^{3} + 14 T^{4} - 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 12 T^{3} - 178 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 324 T^{3} + 1262 T^{4} + 324 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 68 T^{3} + 574 T^{4} + 68 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 646 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_4\times C_2$ | \( 1 + 28 T^{2} + 966 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 92 T^{3} + 862 T^{4} + 92 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4$ | \( ( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 116 T^{3} + 1486 T^{4} + 116 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 660 T^{3} + 6046 T^{4} - 660 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 - 82 T^{2} + p^{2} T^{4} ) \) |
| 61 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 236 T^{3} + 6958 T^{4} - 236 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 14086 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 444 T^{3} - 12994 T^{4} + 444 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
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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
−8.767734310781057890338096985306, −7.80146935661512625280145224448, −7.63280501647581216112370897276, −7.53897129773797087448189182808, −7.05146736209596164404617231508, −6.87818205754557574638725320696, −6.60003522144711863808979598495, −6.49391578654705675872044955266, −6.30362131573968834640199601759, −5.93641762778639596123673914671, −5.82246070762017210792379448642, −5.39640746288363285911023579244, −5.22768031196217088281807351875, −4.89844430213316792687765918313, −4.68929831830903392125508341146, −4.26256575707041920285728388327, −4.14544776281565325000678378912, −4.03915799548784227572736964501, −3.30363204555263239169880498292, −3.06594872541961101916096390310, −2.73925746450874589906940968989, −2.45204890383109105561336242009, −2.23517625318762102262077562984, −1.89418719046149637377031316571, −0.74338045163897465537553786839,
0.74338045163897465537553786839, 1.89418719046149637377031316571, 2.23517625318762102262077562984, 2.45204890383109105561336242009, 2.73925746450874589906940968989, 3.06594872541961101916096390310, 3.30363204555263239169880498292, 4.03915799548784227572736964501, 4.14544776281565325000678378912, 4.26256575707041920285728388327, 4.68929831830903392125508341146, 4.89844430213316792687765918313, 5.22768031196217088281807351875, 5.39640746288363285911023579244, 5.82246070762017210792379448642, 5.93641762778639596123673914671, 6.30362131573968834640199601759, 6.49391578654705675872044955266, 6.60003522144711863808979598495, 6.87818205754557574638725320696, 7.05146736209596164404617231508, 7.53897129773797087448189182808, 7.63280501647581216112370897276, 7.80146935661512625280145224448, 8.767734310781057890338096985306
