| L(s) = 1 | − 4·2-s − 4·3-s + 10·4-s + 16·6-s − 20·8-s + 10·9-s − 4·11-s − 40·12-s + 35·16-s − 40·18-s + 16·22-s − 8·23-s + 80·24-s − 4·25-s − 20·27-s − 16·31-s − 56·32-s + 16·33-s + 100·36-s + 8·37-s + 8·43-s − 40·44-s + 32·46-s − 16·47-s − 140·48-s + 16·50-s + 8·53-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 5·4-s + 6.53·6-s − 7.07·8-s + 10/3·9-s − 1.20·11-s − 11.5·12-s + 35/4·16-s − 9.42·18-s + 3.41·22-s − 1.66·23-s + 16.3·24-s − 4/5·25-s − 3.84·27-s − 2.87·31-s − 9.89·32-s + 2.78·33-s + 50/3·36-s + 1.31·37-s + 1.21·43-s − 6.03·44-s + 4.71·46-s − 2.33·47-s − 20.2·48-s + 2.26·50-s + 1.09·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 3 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 + 4 T^{2} + 46 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T^{2} + 32 T^{3} + 242 T^{4} + 32 p T^{5} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 + 36 T^{2} - 32 T^{3} + 638 T^{4} - 32 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 40 T^{2} - 80 T^{3} + 794 T^{4} - 80 p T^{5} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 36 T^{2} - 56 T^{3} - 570 T^{4} - 56 p T^{5} + 36 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 200 T^{2} + 1568 T^{3} + 10378 T^{4} + 1568 p T^{5} + 200 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 76 T^{2} - 408 T^{3} + 2486 T^{4} - 408 p T^{5} + 76 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 132 T^{2} + 32 T^{3} + 7454 T^{4} + 32 p T^{5} + 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 116 T^{2} - 424 T^{3} + 5110 T^{4} - 424 p T^{5} + 116 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 248 T^{2} + 2304 T^{3} + 18890 T^{4} + 2304 p T^{5} + 248 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 140 T^{2} - 1048 T^{3} + 9334 T^{4} - 1048 p T^{5} + 140 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 304 T^{2} + 2864 T^{3} + 29362 T^{4} + 2864 p T^{5} + 304 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 124 T^{2} - 1104 T^{3} + 11510 T^{4} - 1104 p T^{5} + 124 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 + 140 T^{2} - 256 T^{3} + 12422 T^{4} - 256 p T^{5} + 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 116 T^{2} + 448 T^{3} + 6462 T^{4} + 448 p T^{5} + 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 236 T^{2} - 2448 T^{3} + 24582 T^{4} - 2448 p T^{5} + 236 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 280 T^{2} + 48 T^{3} + 32794 T^{4} + 48 p T^{5} + 280 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 160 T^{2} + 1904 T^{3} + 24642 T^{4} + 1904 p T^{5} + 160 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 352 T^{2} - 3984 T^{3} + 51458 T^{4} - 3984 p T^{5} + 352 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.56082032331281953212474100054, −6.24811511635054550795146443622, −6.13379320798146501709413042451, −6.11024177190494655752335378466, −5.87477844325301664105986250875, −5.58518267755050970873880849519, −5.52487197537645071332289976333, −5.26890488185587004830988381720, −5.20430443150634490434156029309, −4.69769880208904966411307553228, −4.48559902733692913720542326892, −4.44854191917653877696469051060, −4.22506924410833323385636052707, −3.67334325864172947751664375050, −3.44954761732885093838302747858, −3.44840438050793502441983074023, −3.24276316508585736914668113187, −2.50362139176396811487760500654, −2.46656731452807846204838169545, −2.22264811937197552883031382220, −2.13439184238392726505370426224, −1.51339317206990527609371692587, −1.51036552432221618052807062560, −1.09878673122210704234196497891, −1.06070368793872061043693060970, 0, 0, 0, 0,
1.06070368793872061043693060970, 1.09878673122210704234196497891, 1.51036552432221618052807062560, 1.51339317206990527609371692587, 2.13439184238392726505370426224, 2.22264811937197552883031382220, 2.46656731452807846204838169545, 2.50362139176396811487760500654, 3.24276316508585736914668113187, 3.44840438050793502441983074023, 3.44954761732885093838302747858, 3.67334325864172947751664375050, 4.22506924410833323385636052707, 4.44854191917653877696469051060, 4.48559902733692913720542326892, 4.69769880208904966411307553228, 5.20430443150634490434156029309, 5.26890488185587004830988381720, 5.52487197537645071332289976333, 5.58518267755050970873880849519, 5.87477844325301664105986250875, 6.11024177190494655752335378466, 6.13379320798146501709413042451, 6.24811511635054550795146443622, 6.56082032331281953212474100054
