| L(s) = 1 | + 106·4-s + 586·9-s − 1.23e3·11-s + 4.84e3·16-s + 6.72e3·19-s − 4.61e3·29-s − 1.27e4·31-s + 6.21e4·36-s + 1.65e4·41-s − 1.30e5·44-s + 3.42e4·49-s − 9.50e3·59-s − 1.73e5·61-s + 8.52e4·64-s − 2.46e5·71-s + 7.12e5·76-s + 2.21e5·79-s + 7.82e4·81-s + 2.24e5·89-s − 7.21e5·99-s + 6.09e4·101-s − 6.08e5·109-s − 4.88e5·116-s − 1.51e4·121-s − 1.35e6·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 3.31·4-s + 2.41·9-s − 3.06·11-s + 4.72·16-s + 4.27·19-s − 1.01·29-s − 2.38·31-s + 7.98·36-s + 1.53·41-s − 10.1·44-s + 2.03·49-s − 0.355·59-s − 5.97·61-s + 2.60·64-s − 5.80·71-s + 14.1·76-s + 3.99·79-s + 1.32·81-s + 3.00·89-s − 7.40·99-s + 0.594·101-s − 4.90·109-s − 3.37·116-s − 0.0943·121-s − 7.89·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.170844367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.170844367\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( ( 1 + p^{4} T^{2} )^{3} \) |
| good | 2 | \( 1 - 53 p T^{2} + 6393 T^{4} - 62375 p^{2} T^{6} + 6393 p^{10} T^{8} - 53 p^{21} T^{10} + p^{30} T^{12} \) |
| 3 | \( 1 - 586 T^{2} + 265111 T^{4} - 7921772 p^{2} T^{6} + 265111 p^{10} T^{8} - 586 p^{20} T^{10} + p^{30} T^{12} \) |
| 7 | \( 1 - 34250 T^{2} + 1157349791 T^{4} - 19859968898444 T^{6} + 1157349791 p^{10} T^{8} - 34250 p^{20} T^{10} + p^{30} T^{12} \) |
| 11 | \( ( 1 + 56 p T + 576781 T^{2} + 198710800 T^{3} + 576781 p^{5} T^{4} + 56 p^{11} T^{5} + p^{15} T^{6} )^{2} \) |
| 17 | \( 1 - 5346778 T^{2} + 14107385735535 T^{4} - 24198875660122621868 T^{6} + 14107385735535 p^{10} T^{8} - 5346778 p^{20} T^{10} + p^{30} T^{12} \) |
| 19 | \( ( 1 - 3360 T + 9980613 T^{2} - 16920687808 T^{3} + 9980613 p^{5} T^{4} - 3360 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 23 | \( 1 - 17947666 T^{2} + 177782686483887 T^{4} - \)\(12\!\cdots\!92\)\( T^{6} + 177782686483887 p^{10} T^{8} - 17947666 p^{20} T^{10} + p^{30} T^{12} \) |
| 29 | \( ( 1 + 2306 T + 63301891 T^{2} + 95048382380 T^{3} + 63301891 p^{5} T^{4} + 2306 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 31 | \( ( 1 + 6380 T + 63602873 T^{2} + 216555662264 T^{3} + 63602873 p^{5} T^{4} + 6380 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 37 | \( 1 - 196213586 T^{2} + 17914233453773591 T^{4} - \)\(12\!\cdots\!88\)\( T^{6} + 17914233453773591 p^{10} T^{8} - 196213586 p^{20} T^{10} + p^{30} T^{12} \) |
| 41 | \( ( 1 - 8286 T + 210623751 T^{2} - 2260028786820 T^{3} + 210623751 p^{5} T^{4} - 8286 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 43 | \( 1 - 639846330 T^{2} + 198848638114826151 T^{4} - \)\(36\!\cdots\!56\)\( T^{6} + 198848638114826151 p^{10} T^{8} - 639846330 p^{20} T^{10} + p^{30} T^{12} \) |
| 47 | \( 1 - 1108811578 T^{2} + 563976871326771567 T^{4} - \)\(16\!\cdots\!80\)\( T^{6} + 563976871326771567 p^{10} T^{8} - 1108811578 p^{20} T^{10} + p^{30} T^{12} \) |
| 53 | \( 1 - 424381554 T^{2} + 113706106857181431 T^{4} - \)\(92\!\cdots\!52\)\( T^{6} + 113706106857181431 p^{10} T^{8} - 424381554 p^{20} T^{10} + p^{30} T^{12} \) |
| 59 | \( ( 1 + 4752 T + 763206237 T^{2} + 23933901228768 T^{3} + 763206237 p^{5} T^{4} + 4752 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 61 | \( ( 1 + 86894 T + 3218447267 T^{2} + 88402029353204 T^{3} + 3218447267 p^{5} T^{4} + 86894 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 67 | \( 1 + 931485438 T^{2} + 2776490132958986535 T^{4} + \)\(14\!\cdots\!36\)\( T^{6} + 2776490132958986535 p^{10} T^{8} + 931485438 p^{20} T^{10} + p^{30} T^{12} \) |
| 71 | \( ( 1 + 123252 T + 7672133745 T^{2} + 347878540025160 T^{3} + 7672133745 p^{5} T^{4} + 123252 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 73 | \( 1 - 6548382410 T^{2} + 25607390713292226047 T^{4} - \)\(63\!\cdots\!80\)\( T^{6} + 25607390713292226047 p^{10} T^{8} - 6548382410 p^{20} T^{10} + p^{30} T^{12} \) |
| 79 | \( ( 1 - 110696 T + 10847564381 T^{2} - 593275027409072 T^{3} + 10847564381 p^{5} T^{4} - 110696 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 83 | \( 1 - 16170291778 T^{2} + \)\(12\!\cdots\!15\)\( T^{4} - \)\(61\!\cdots\!64\)\( T^{6} + \)\(12\!\cdots\!15\)\( p^{10} T^{8} - 16170291778 p^{20} T^{10} + p^{30} T^{12} \) |
| 89 | \( ( 1 - 112210 T + 13708842775 T^{2} - 819892069403740 T^{3} + 13708842775 p^{5} T^{4} - 112210 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 97 | \( 1 - 29962994618 T^{2} + \)\(40\!\cdots\!79\)\( T^{4} - \)\(37\!\cdots\!28\)\( T^{6} + \)\(40\!\cdots\!79\)\( p^{10} T^{8} - 29962994618 p^{20} T^{10} + p^{30} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.37853881216288424252125835570, −5.36248492467707382710155723171, −5.23113297508819732677710988640, −4.98317167694896253399841079879, −4.65009716430432302613886380304, −4.42326549095289092694903424622, −4.41416057933997276156081313444, −3.89358309487180517708316339660, −3.78214763469832472882748376347, −3.76098703836354212517163431541, −3.16697511899316000262628465123, −2.99868734844195749244487282668, −2.97031795652319445311268804565, −2.78728132722754434436462039183, −2.63213673373330264577622190552, −2.59439118495206476384264718433, −1.97305676170305622553623191302, −1.83107112656120550957677172220, −1.74781311710972457969136815359, −1.49729564351410859940226877168, −1.41622499796912474082293003719, −1.17904378966520690069092343914, −0.69225113690646767743213188588, −0.50602691590769484350880232678, −0.13160402507789351749944687784,
0.13160402507789351749944687784, 0.50602691590769484350880232678, 0.69225113690646767743213188588, 1.17904378966520690069092343914, 1.41622499796912474082293003719, 1.49729564351410859940226877168, 1.74781311710972457969136815359, 1.83107112656120550957677172220, 1.97305676170305622553623191302, 2.59439118495206476384264718433, 2.63213673373330264577622190552, 2.78728132722754434436462039183, 2.97031795652319445311268804565, 2.99868734844195749244487282668, 3.16697511899316000262628465123, 3.76098703836354212517163431541, 3.78214763469832472882748376347, 3.89358309487180517708316339660, 4.41416057933997276156081313444, 4.42326549095289092694903424622, 4.65009716430432302613886380304, 4.98317167694896253399841079879, 5.23113297508819732677710988640, 5.36248492467707382710155723171, 5.37853881216288424252125835570
Plot not available for L-functions of degree greater than 10.
