| L(s) = 1 | + 48·4-s − 40·5-s + 54·9-s + 1.26e3·16-s − 1.92e3·20-s − 1.22e3·23-s + 444·25-s − 2.48e3·31-s + 2.59e3·36-s − 304·37-s − 2.16e3·45-s − 6.37e3·47-s + 8.35e3·49-s + 9.40e3·53-s − 1.13e4·59-s + 2.30e4·64-s − 64·67-s + 5.43e3·71-s − 5.05e4·80-s + 2.18e3·81-s + 2.04e4·89-s − 5.87e4·92-s + 4.37e4·97-s + 2.13e4·100-s − 3.57e4·103-s + 344·113-s + 4.89e4·115-s + ⋯ |
| L(s) = 1 | + 3·4-s − 8/5·5-s + 2/3·9-s + 4.93·16-s − 4.79·20-s − 2.31·23-s + 0.710·25-s − 2.58·31-s + 2·36-s − 0.222·37-s − 1.06·45-s − 2.88·47-s + 3.47·49-s + 3.34·53-s − 3.26·59-s + 45/8·64-s − 0.0142·67-s + 1.07·71-s − 7.89·80-s + 1/3·81-s + 2.57·89-s − 6.94·92-s + 4.64·97-s + 2.13·100-s − 3.36·103-s + 0.0269·113-s + 3.70·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.517905776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.517905776\) |
| \(L(3)\) |
|
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 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 3 p^{4} T^{2} + 65 p^{4} T^{4} - 3 p^{12} T^{6} + p^{16} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 + 4 p T + 378 T^{2} + 4 p^{5} T^{3} + p^{8} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 8352 T^{2} + 28785551 T^{4} - 8352 p^{8} T^{6} + p^{16} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 63760 T^{2} + 2588655042 T^{4} - 63760 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 24628 T^{2} + 9014783910 T^{4} - 24628 p^{8} T^{6} + p^{16} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 510976 T^{2} + 99214834959 T^{4} - 510976 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 612 T + 368726 T^{2} + 612 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 416052 T^{2} + 67384276406 T^{4} - 416052 p^{8} T^{6} + p^{16} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 40 p T + 2223015 T^{2} + 40 p^{5} T^{3} + p^{8} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 152 T - 1512777 T^{2} + 152 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 3471700 T^{2} + 10422795219894 T^{4} - 3471700 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 544880 T^{2} + 21042735180354 T^{4} + 544880 p^{8} T^{6} + p^{16} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 3188 T + 11191206 T^{2} + 3188 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 4700 T + 14002662 T^{2} - 4700 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 5680 T + 23112822 T^{2} + 5680 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 30176832 T^{2} + 588010079781743 T^{4} - 30176832 p^{8} T^{6} + p^{16} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 32 T - 6291945 T^{2} + 32 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 2716 T + 45235098 T^{2} - 2716 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 103855152 T^{2} + 4285811216064863 T^{4} - 103855152 p^{8} T^{6} + p^{16} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 133033552 T^{2} + 7330387723104591 T^{4} - 133033552 p^{8} T^{6} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 25060884 T^{2} + 3293116763763446 T^{4} - 25060884 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 10200 T + 124422434 T^{2} - 10200 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 21866 T + 281374539 T^{2} - 21866 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
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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
−7.53801615744033965405651893231, −7.36211783572902655968118421339, −7.22540446857719407841775543511, −6.97037285978456427298644128383, −6.78150750096090135964334973749, −6.24738068425522464072326839315, −6.20785899956884145727159800760, −6.04234959300653113540962305106, −5.71908029250122266029873465981, −5.32664199096965779511823124490, −5.15173350947988702625810658443, −4.54669725329342117101559899843, −4.45681403112650817516672252524, −3.78971724860352766476090195851, −3.75577876228898884674858434336, −3.59713942569644262280563767234, −3.35272699545164717171966485357, −2.79908518448671880489080292202, −2.27298637423711357717523896175, −2.24616421960202689134744301716, −2.05228362062410842578205413299, −1.50161035593782749620860117661, −1.30700777624260287899447961489, −0.66687283160506218353151062076, −0.18997166768054961263553388248,
0.18997166768054961263553388248, 0.66687283160506218353151062076, 1.30700777624260287899447961489, 1.50161035593782749620860117661, 2.05228362062410842578205413299, 2.24616421960202689134744301716, 2.27298637423711357717523896175, 2.79908518448671880489080292202, 3.35272699545164717171966485357, 3.59713942569644262280563767234, 3.75577876228898884674858434336, 3.78971724860352766476090195851, 4.45681403112650817516672252524, 4.54669725329342117101559899843, 5.15173350947988702625810658443, 5.32664199096965779511823124490, 5.71908029250122266029873465981, 6.04234959300653113540962305106, 6.20785899956884145727159800760, 6.24738068425522464072326839315, 6.78150750096090135964334973749, 6.97037285978456427298644128383, 7.22540446857719407841775543511, 7.36211783572902655968118421339, 7.53801615744033965405651893231
