| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 4·5-s + 2·6-s + 3·7-s − 5·8-s − 8·10-s − 2·12-s − 13-s − 6·14-s − 4·15-s + 5·16-s + 3·17-s − 5·19-s + 8·20-s − 3·21-s − 4·23-s + 5·24-s + 15·25-s + 2·26-s + 6·28-s − 10·29-s + 8·30-s − 7·31-s + 2·32-s − 6·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 1.78·5-s + 0.816·6-s + 1.13·7-s − 1.76·8-s − 2.52·10-s − 0.577·12-s − 0.277·13-s − 1.60·14-s − 1.03·15-s + 5/4·16-s + 0.727·17-s − 1.14·19-s + 1.78·20-s − 0.654·21-s − 0.834·23-s + 1.02·24-s + 3·25-s + 0.392·26-s + 1.13·28-s − 1.85·29-s + 1.46·30-s − 1.25·31-s + 0.353·32-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.095321145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.095321145\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2:C_4$ | \( 1 + p T + p T^{2} + 5 T^{3} + 11 T^{4} + 5 p T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 - 4 T + T^{2} + 16 T^{3} - 39 T^{4} + 16 p T^{5} + p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - 3 T + 2 T^{2} + 15 T^{3} - 59 T^{4} + 15 p T^{5} + 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + T + 18 T^{2} + 5 T^{3} + 251 T^{4} + 5 p T^{5} + 18 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 3 T - 13 T^{2} + 15 T^{3} + 256 T^{4} + 15 p T^{5} - 13 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 5 T + 21 T^{2} + 145 T^{3} + 956 T^{4} + 145 p T^{5} + 21 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4\times C_2$ | \( 1 + 10 T + 31 T^{2} + 200 T^{3} + 1821 T^{4} + 200 p T^{5} + 31 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 7 T + 3 T^{2} + 119 T^{3} + 1640 T^{4} + 119 p T^{5} + 3 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 7 T - 18 T^{2} + 145 T^{3} + 371 T^{4} + 145 p T^{5} - 18 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 17 T + 208 T^{2} - 1879 T^{3} + 13815 T^{4} - 1879 p T^{5} + 208 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 2 T - 43 T^{2} - 50 T^{3} + 2351 T^{4} - 50 p T^{5} - 43 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 11 T + 23 T^{2} + 475 T^{3} - 5824 T^{4} + 475 p T^{5} + 23 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 15 T + 131 T^{2} - 1395 T^{3} + 14176 T^{4} - 1395 p T^{5} + 131 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 12 T - 7 T^{2} + 576 T^{3} - 3335 T^{4} + 576 p T^{5} - 7 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 18 T + 253 T^{2} - 2826 T^{3} + 28855 T^{4} - 2826 p T^{5} + 253 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 4 T - 57 T^{2} - 170 T^{3} + 6221 T^{4} - 170 p T^{5} - 57 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 15 T + 6 T^{2} + 815 T^{3} - 5979 T^{4} + 815 p T^{5} + 6 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 9 T + 88 T^{2} - 1185 T^{3} + 16681 T^{4} - 1185 p T^{5} + 88 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 10 T + 183 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 32 T + 537 T^{2} - 7060 T^{3} + 77501 T^{4} - 7060 p T^{5} + 537 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.333120030424266702211878757052, −8.157925404452610575249318927603, −8.110084534853654804389010326276, −7.38260595263824312404188892283, −7.29120252749070145096650346640, −7.10175201479451955716167013271, −6.99193715476250176634363956813, −6.19671659381611301634911759590, −6.14463405279921514306950701578, −6.12542947214131653677647841175, −6.08403851355427649296377772560, −5.37469693186460567870880355434, −5.25170572820194156141639314317, −4.97070354680024222411578640611, −4.94678110410953670760237073959, −4.26101368014768790561735425307, −3.94664202589201957480362358997, −3.50395532119027538868892816777, −3.38454046814789050640293100230, −2.53850750597214293231704952973, −2.19377068025384924919431543777, −2.13314915279618859584762450438, −2.02375002024863075118287765577, −0.928664665859263702013630722984, −0.71691118750311535038042708788,
0.71691118750311535038042708788, 0.928664665859263702013630722984, 2.02375002024863075118287765577, 2.13314915279618859584762450438, 2.19377068025384924919431543777, 2.53850750597214293231704952973, 3.38454046814789050640293100230, 3.50395532119027538868892816777, 3.94664202589201957480362358997, 4.26101368014768790561735425307, 4.94678110410953670760237073959, 4.97070354680024222411578640611, 5.25170572820194156141639314317, 5.37469693186460567870880355434, 6.08403851355427649296377772560, 6.12542947214131653677647841175, 6.14463405279921514306950701578, 6.19671659381611301634911759590, 6.99193715476250176634363956813, 7.10175201479451955716167013271, 7.29120252749070145096650346640, 7.38260595263824312404188892283, 8.110084534853654804389010326276, 8.157925404452610575249318927603, 8.333120030424266702211878757052
