| L(s) = 1 | − 16·9-s + 208·13-s − 312·29-s + 272·37-s − 96·41-s − 80·49-s + 48·53-s + 1.05e3·61-s + 1.44e3·73-s − 510·81-s − 40·89-s + 4.54e3·97-s − 1.76e3·101-s − 1.69e3·109-s + 64·113-s − 3.32e3·117-s − 332·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.89e4·169-s + ⋯ |
| L(s) = 1 | − 0.592·9-s + 4.43·13-s − 1.99·29-s + 1.20·37-s − 0.365·41-s − 0.233·49-s + 0.124·53-s + 2.21·61-s + 2.30·73-s − 0.699·81-s − 0.0476·89-s + 4.75·97-s − 1.74·101-s − 1.49·109-s + 0.0532·113-s − 2.62·117-s − 0.249·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 8.61·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.488692020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.488692020\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 + 16 T^{2} + 766 T^{4} + 16 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 80 T^{2} + 212622 T^{4} + 80 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 332 T^{2} + 474102 T^{4} + 332 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 8 p T + 6762 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 4450 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 2644 T^{2} + 87237846 T^{4} - 2644 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 36816 T^{2} + 602323406 T^{4} + 36816 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 156 T + 33358 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 95612 T^{2} + 4010875782 T^{4} + 95612 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 136 T + 2418 p T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 48 T + 124222 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 66320 T^{2} - 2039763618 T^{4} + 66320 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 204240 T^{2} + 24337141742 T^{4} + 204240 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 24 T + 222298 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 507788 T^{2} + 128506595382 T^{4} + 507788 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 528 T + 462422 T^{2} - 528 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 814480 T^{2} + 336078840702 T^{4} + 814480 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 1385052 T^{2} + 735523082342 T^{4} + 1385052 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 720 T + 745010 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 756668 T^{2} + 456844629702 T^{4} + 756668 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 300176 T^{2} + 429308372286 T^{4} + 300176 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 20 T + 872438 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1136 T + p^{3} T^{2} )^{4} \) |
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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
−7.02046360750259610842801305105, −6.51323909273190285989106661552, −6.43783341467725091029213986642, −6.24895349737576471200378165225, −6.21089296966563334438438567013, −5.69441452532037748975409747220, −5.51314532281432818786067571164, −5.48872866045405493505686102011, −5.37621223275048865029826068129, −4.67112808621474684747770832593, −4.55819496858862933857354727117, −4.26129865965972594680182191796, −3.90405185047841003700219860274, −3.72340125895862015426569334152, −3.45774296083016610866741811636, −3.32133230128389969787543840573, −3.30970310313647368586941338858, −2.51676929386042629931610906622, −2.30255122758277711542873347029, −2.06064059291034225813135847268, −1.64333868608935314042565762547, −1.17173528145710242574822823007, −1.04093935620705369036404150686, −0.841506640365310097175523739524, −0.18231366634640298327123630803,
0.18231366634640298327123630803, 0.841506640365310097175523739524, 1.04093935620705369036404150686, 1.17173528145710242574822823007, 1.64333868608935314042565762547, 2.06064059291034225813135847268, 2.30255122758277711542873347029, 2.51676929386042629931610906622, 3.30970310313647368586941338858, 3.32133230128389969787543840573, 3.45774296083016610866741811636, 3.72340125895862015426569334152, 3.90405185047841003700219860274, 4.26129865965972594680182191796, 4.55819496858862933857354727117, 4.67112808621474684747770832593, 5.37621223275048865029826068129, 5.48872866045405493505686102011, 5.51314532281432818786067571164, 5.69441452532037748975409747220, 6.21089296966563334438438567013, 6.24895349737576471200378165225, 6.43783341467725091029213986642, 6.51323909273190285989106661552, 7.02046360750259610842801305105
