| L(s) = 1 | − 2·2-s − 6·3-s + 2·4-s + 2·5-s + 12·6-s + 8·7-s − 4·8-s + 21·9-s − 4·10-s + 4·11-s − 12·12-s − 16·14-s − 12·15-s + 8·16-s + 12·17-s − 42·18-s − 12·19-s + 4·20-s − 48·21-s − 8·22-s + 24·24-s + 5·25-s − 54·27-s + 16·28-s + 18·29-s + 24·30-s + 6·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 3.46·3-s + 4-s + 0.894·5-s + 4.89·6-s + 3.02·7-s − 1.41·8-s + 7·9-s − 1.26·10-s + 1.20·11-s − 3.46·12-s − 4.27·14-s − 3.09·15-s + 2·16-s + 2.91·17-s − 9.89·18-s − 2.75·19-s + 0.894·20-s − 10.4·21-s − 1.70·22-s + 4.89·24-s + 25-s − 10.3·27-s + 3.02·28-s + 3.34·29-s + 4.38·30-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6920996267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6920996267\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2$$\times$$C_2^2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 36 T^{2} - 12 T^{3} + 599 T^{4} - 12 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 9 T^{2} + 120 T^{3} - 244 T^{4} + 120 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 18 T + 189 T^{2} - 1458 T^{3} + 8852 T^{4} - 1458 p T^{5} + 189 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 6 T - 8 T^{2} + 108 T^{3} - 141 T^{4} + 108 p T^{5} - 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 372 T^{3} + 1886 T^{4} - 372 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 12 T + 133 T^{2} + 1020 T^{3} + 7512 T^{4} + 1020 p T^{5} + 133 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6 T + 90 T^{2} - 624 T^{3} + 4895 T^{4} - 624 p T^{5} + 90 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 6 T + 45 T^{2} + 390 T^{3} + 1256 T^{4} + 390 p T^{5} + 45 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 2914 T^{4} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 18 T + 204 T^{2} + 1728 T^{3} + 12107 T^{4} + 1728 p T^{5} + 204 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $D_4\times C_2$ | \( 1 + 6 T + 45 T^{2} - 738 T^{3} - 4576 T^{4} - 738 p T^{5} + 45 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 1380 T^{3} + 9506 T^{4} + 1380 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 18 T + 284 T^{2} - 3168 T^{3} + 33267 T^{4} - 3168 p T^{5} + 284 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 28 T + 365 T^{2} + 2980 T^{3} + 23356 T^{4} + 2980 p T^{5} + 365 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 139 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
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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
−8.329337473634452127190702973499, −8.311553079640436439186212539138, −7.913883925498489434741448411608, −7.52668133071366161070909193408, −7.20478801584072286510972226365, −6.84077294627299122374277312122, −6.78657717347621211806428700104, −6.56228541637883931587248145030, −5.98720574934315442036585138748, −5.97056494127669404126114920314, −5.93941032462266938949121957462, −5.69545457076010114832811958796, −5.16284364332135599421770305333, −4.96383591747260510959626584049, −4.66381446021428820949269214582, −4.55120165957053361353139501838, −4.35079377124842871626522163057, −4.00736884418332958760259752986, −3.06568392579356641276332740867, −3.01046402797399078893049309219, −2.14773375472154076769518389454, −1.60655476896511758223169722269, −1.47594846873974213564243768243, −0.888346610320042094988748355639, −0.874667160316508088734724658460,
0.874667160316508088734724658460, 0.888346610320042094988748355639, 1.47594846873974213564243768243, 1.60655476896511758223169722269, 2.14773375472154076769518389454, 3.01046402797399078893049309219, 3.06568392579356641276332740867, 4.00736884418332958760259752986, 4.35079377124842871626522163057, 4.55120165957053361353139501838, 4.66381446021428820949269214582, 4.96383591747260510959626584049, 5.16284364332135599421770305333, 5.69545457076010114832811958796, 5.93941032462266938949121957462, 5.97056494127669404126114920314, 5.98720574934315442036585138748, 6.56228541637883931587248145030, 6.78657717347621211806428700104, 6.84077294627299122374277312122, 7.20478801584072286510972226365, 7.52668133071366161070909193408, 7.913883925498489434741448411608, 8.311553079640436439186212539138, 8.329337473634452127190702973499
