| L(s) = 1 | + 364·7-s − 616·13-s − 8.38e3·19-s + 2.47e4·25-s − 1.05e4·31-s − 2.71e4·37-s + 5.98e4·43-s − 6.11e4·49-s − 7.97e5·61-s − 4.26e5·67-s + 7.35e5·73-s + 2.30e6·79-s − 2.24e5·91-s − 4.60e6·97-s − 3.77e6·103-s + 1.29e6·109-s + 6.76e6·121-s + 127-s + 131-s − 3.05e6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.06·7-s − 0.280·13-s − 1.22·19-s + 1.58·25-s − 0.354·31-s − 0.536·37-s + 0.753·43-s − 0.520·49-s − 3.51·61-s − 1.41·67-s + 1.88·73-s + 4.66·79-s − 0.297·91-s − 5.04·97-s − 3.45·103-s + 1.00·109-s + 3.81·121-s − 1.29·133-s − 3.68·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.325153931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.325153931\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 24754 T^{2} + 23542539 p^{2} T^{4} - 24754 p^{12} T^{6} + p^{24} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 26 p T + 11469 p T^{2} - 26 p^{7} T^{3} + p^{12} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 6764674 T^{2} + 17691452909475 T^{4} - 6764674 p^{12} T^{6} + p^{24} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 308 T + 9024150 T^{2} + 308 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 47799292 T^{2} + 1158872863632774 T^{4} - 47799292 p^{12} T^{6} + p^{24} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 4192 T + 19449714 T^{2} + 4192 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 18534236 T^{2} + 5801036172284166 T^{4} + 18534236 p^{12} T^{6} + p^{24} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 125082076 T^{2} - 19167490066906074 T^{4} - 125082076 p^{12} T^{6} + p^{24} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 5278 T + 1768744707 T^{2} + 5278 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 13592 T + 2417936034 T^{2} + 13592 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2221568932 T^{2} + 44327483009987910918 T^{4} - 2221568932 p^{12} T^{6} + p^{24} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 29948 T + 8581406550 T^{2} - 29948 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 19885993276 T^{2} + \)\(23\!\cdots\!10\)\( T^{4} - 19885993276 p^{12} T^{6} + p^{24} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 32810027026 T^{2} + \)\(48\!\cdots\!75\)\( T^{4} - 32810027026 p^{12} T^{6} + p^{24} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 491131804 T^{2} - \)\(99\!\cdots\!70\)\( T^{4} - 491131804 p^{12} T^{6} + p^{24} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 398936 T + 126932345922 T^{2} + 398936 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 213388 T + 100446373974 T^{2} + 213388 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 432605134588 T^{2} + \)\(78\!\cdots\!14\)\( T^{4} - 432605134588 p^{12} T^{6} + p^{24} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 367522 T + 305109199875 T^{2} - 367522 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 1151012 T + 790729547142 T^{2} - 1151012 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 199657697458 T^{2} + \)\(61\!\cdots\!63\)\( T^{4} - 199657697458 p^{12} T^{6} + p^{24} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 1636569283324 T^{2} + \)\(11\!\cdots\!10\)\( T^{4} - 1636569283324 p^{12} T^{6} + p^{24} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 2301482 T + 2655290162235 T^{2} + 2301482 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
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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
−7.20349152533646653626548359147, −6.65614896644060750597846119634, −6.63373775185915900281936596418, −6.34089195583036419257211494582, −6.04085835442621896632057067563, −5.90865892165811115432516155653, −5.33698877899266374722114860457, −5.24692176013485148088861678033, −5.01794410210188028496310209777, −4.79916358876995308132301165177, −4.53421471289291847677740164224, −4.18350551389894830118032923524, −4.01832368825681244898443784300, −3.74400970407500138090667260766, −3.27818177728940258100808274736, −3.03117589801032025578314411113, −2.66219036747019793750369207592, −2.59925469773186173886408060404, −1.97194017998689388223803377900, −1.88935830309192231823691281478, −1.51031715783271750358225381024, −1.25749063884272590048236288247, −0.918741395547107586919857464224, −0.42625274596382903360924877497, −0.23907069881720822398592814887,
0.23907069881720822398592814887, 0.42625274596382903360924877497, 0.918741395547107586919857464224, 1.25749063884272590048236288247, 1.51031715783271750358225381024, 1.88935830309192231823691281478, 1.97194017998689388223803377900, 2.59925469773186173886408060404, 2.66219036747019793750369207592, 3.03117589801032025578314411113, 3.27818177728940258100808274736, 3.74400970407500138090667260766, 4.01832368825681244898443784300, 4.18350551389894830118032923524, 4.53421471289291847677740164224, 4.79916358876995308132301165177, 5.01794410210188028496310209777, 5.24692176013485148088861678033, 5.33698877899266374722114860457, 5.90865892165811115432516155653, 6.04085835442621896632057067563, 6.34089195583036419257211494582, 6.63373775185915900281936596418, 6.65614896644060750597846119634, 7.20349152533646653626548359147
