| L(s) = 1 | + 8·5-s + 16·13-s − 16-s − 8·17-s − 8·23-s + 38·25-s + 8·29-s + 24·31-s + 8·37-s − 4·43-s − 20·47-s + 16·53-s + 128·65-s + 28·67-s − 8·80-s − 8·83-s − 64·85-s − 24·89-s − 24·97-s − 24·103-s + 32·107-s − 64·115-s + 32·121-s + 136·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 3.57·5-s + 4.43·13-s − 1/4·16-s − 1.94·17-s − 1.66·23-s + 38/5·25-s + 1.48·29-s + 4.31·31-s + 1.31·37-s − 0.609·43-s − 2.91·47-s + 2.19·53-s + 15.8·65-s + 3.42·67-s − 0.894·80-s − 0.878·83-s − 6.94·85-s − 2.54·89-s − 2.43·97-s − 2.36·103-s + 3.09·107-s − 5.96·115-s + 2.90·121-s + 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.644275135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.644275135\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 11 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 466 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 168 T^{3} + 866 T^{4} + 168 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 120 T^{3} + 386 T^{4} + 120 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 72 T^{3} - 622 T^{4} - 72 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3654 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} - 220 T^{3} - 3554 T^{4} - 220 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 1220 T^{3} + 7246 T^{4} + 1220 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1296 T^{3} + 12338 T^{4} - 1296 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 28 T + 392 T^{2} - 4508 T^{3} + 43006 T^{4} - 4508 p T^{5} + 392 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 23494 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^3$ | \( 1 - 8542 T^{4} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 600 T^{3} + 11186 T^{4} + 600 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 206 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 3960 T^{3} + 49826 T^{4} + 3960 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.88780434404857044727125624792, −7.01871557084179708342590129189, −6.95213891709120894685722844489, −6.78784412668583983455798591060, −6.33587181587108567790383491496, −6.32908503683765093839126747524, −6.25585868554271363497376521918, −6.19574283482600675327223362301, −5.90873556954933251915006328514, −5.58769872883621679140965803695, −5.33741629323801550905391353227, −4.94533003112060445076883295109, −4.77098550750050741510346876486, −4.37954327752272488647592107559, −4.32005076872884917956944396293, −3.71204865024957000055138410419, −3.67179433546089780236381731162, −3.19811156262013146245833728904, −2.53998568666270445304789557769, −2.53160967454764911143447382111, −2.47693739114040007180565367467, −1.91694527609568091786129412541, −1.25050269908082119073552964225, −1.24802337923093026931684889067, −1.11595192945629647543818780295,
1.11595192945629647543818780295, 1.24802337923093026931684889067, 1.25050269908082119073552964225, 1.91694527609568091786129412541, 2.47693739114040007180565367467, 2.53160967454764911143447382111, 2.53998568666270445304789557769, 3.19811156262013146245833728904, 3.67179433546089780236381731162, 3.71204865024957000055138410419, 4.32005076872884917956944396293, 4.37954327752272488647592107559, 4.77098550750050741510346876486, 4.94533003112060445076883295109, 5.33741629323801550905391353227, 5.58769872883621679140965803695, 5.90873556954933251915006328514, 6.19574283482600675327223362301, 6.25585868554271363497376521918, 6.32908503683765093839126747524, 6.33587181587108567790383491496, 6.78784412668583983455798591060, 6.95213891709120894685722844489, 7.01871557084179708342590129189, 7.88780434404857044727125624792
