| L(s) = 1 | − 4·3-s + 8·9-s + 8·13-s − 24·23-s − 12·27-s + 24·37-s − 32·39-s + 8·47-s + 16·49-s − 16·59-s + 96·69-s + 16·71-s − 40·73-s + 23·81-s + 8·83-s − 8·97-s + 56·107-s + 48·109-s − 96·111-s + 64·117-s − 28·121-s + 127-s + 131-s + 137-s + 139-s − 32·141-s − 64·147-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 8/3·9-s + 2.21·13-s − 5.00·23-s − 2.30·27-s + 3.94·37-s − 5.12·39-s + 1.16·47-s + 16/7·49-s − 2.08·59-s + 11.5·69-s + 1.89·71-s − 4.68·73-s + 23/9·81-s + 0.878·83-s − 0.812·97-s + 5.41·107-s + 4.59·109-s − 9.11·111-s + 5.91·117-s − 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.69·141-s − 5.27·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8473810095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8473810095\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 130 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 17 | $C_4\times C_2$ | \( 1 + 4 T^{2} + 70 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 32 T^{2} + 2386 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 4 T + 80 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 10006 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 4082 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_4$ | \( ( 1 + 20 T + 214 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 12774 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T + 120 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 8806 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_4$ | \( ( 1 + 4 T - 90 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.17713149860678643331383221510, −6.10845405453645328375740149373, −6.03307106001452051527081657612, −5.98754320303236535099207583758, −5.74352270801399392139480721057, −5.38245422755165215201302202957, −5.26374552899452608100256393839, −4.88336318225739096748002627665, −4.77805871966407091685237281692, −4.27482419271497037993503947796, −4.15407567677816896205566677103, −4.15051117965543369882137859568, −4.09639996082715371386178623933, −3.73802989254181979652992466693, −3.43392920585577862370235261903, −3.09404254666291495435056255886, −2.87026187545546930864971226942, −2.41219003025907057732557534086, −2.09894035753398415400069256035, −2.03423143976675607824762267787, −1.70051347992213991493081183780, −1.18943428194336203479094081616, −0.999532709386729663943355767692, −0.63659813908560042971152972401, −0.25946295004829184477291133831,
0.25946295004829184477291133831, 0.63659813908560042971152972401, 0.999532709386729663943355767692, 1.18943428194336203479094081616, 1.70051347992213991493081183780, 2.03423143976675607824762267787, 2.09894035753398415400069256035, 2.41219003025907057732557534086, 2.87026187545546930864971226942, 3.09404254666291495435056255886, 3.43392920585577862370235261903, 3.73802989254181979652992466693, 4.09639996082715371386178623933, 4.15051117965543369882137859568, 4.15407567677816896205566677103, 4.27482419271497037993503947796, 4.77805871966407091685237281692, 4.88336318225739096748002627665, 5.26374552899452608100256393839, 5.38245422755165215201302202957, 5.74352270801399392139480721057, 5.98754320303236535099207583758, 6.03307106001452051527081657612, 6.10845405453645328375740149373, 6.17713149860678643331383221510
