| L(s) = 1 | − 16·4-s + 240·9-s − 388·11-s − 916·16-s − 2.42e3·19-s + 2.26e4·29-s + 1.84e4·31-s − 3.84e3·36-s + 2.08e4·41-s + 6.20e3·44-s − 7.20e3·49-s − 4.75e4·59-s + 1.74e4·61-s + 3.03e4·64-s − 1.05e5·71-s + 3.87e4·76-s + 1.43e5·79-s − 3.25e4·81-s + 3.85e5·89-s − 9.31e4·99-s + 1.08e5·101-s − 9.92e5·109-s − 3.62e5·116-s − 8.22e5·121-s − 2.94e5·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.987·9-s − 0.966·11-s − 0.894·16-s − 1.54·19-s + 5.00·29-s + 3.43·31-s − 0.493·36-s + 1.94·41-s + 0.483·44-s − 3/7·49-s − 1.77·59-s + 0.600·61-s + 0.925·64-s − 2.48·71-s + 0.770·76-s + 2.57·79-s − 0.551·81-s + 5.15·89-s − 0.954·99-s + 1.05·101-s − 8.00·109-s − 2.50·116-s − 5.10·121-s − 1.71·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.240741366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.240741366\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( ( 1 + p^{4} T^{2} )^{3} \) |
| good | 2 | \( 1 + p^{4} T^{2} + 293 p^{2} T^{4} + 193 p^{4} T^{6} + 293 p^{12} T^{8} + p^{24} T^{10} + p^{30} T^{12} \) |
| 3 | \( 1 - 80 p T^{2} + 30064 p T^{4} - 7512470 T^{6} + 30064 p^{11} T^{8} - 80 p^{21} T^{10} + p^{30} T^{12} \) |
| 11 | \( ( 1 + 194 T + 467570 T^{2} + 59343320 T^{3} + 467570 p^{5} T^{4} + 194 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 13 | \( 1 - 823096 T^{2} + 517034947752 T^{4} - 230107548091734278 T^{6} + 517034947752 p^{10} T^{8} - 823096 p^{20} T^{10} + p^{30} T^{12} \) |
| 17 | \( 1 - 6692080 T^{2} + 20618298560592 T^{4} - 37274918493619967390 T^{6} + 20618298560592 p^{10} T^{8} - 6692080 p^{20} T^{10} + p^{30} T^{12} \) |
| 19 | \( ( 1 + 1212 T + 5059117 T^{2} + 6142323016 T^{3} + 5059117 p^{5} T^{4} + 1212 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 23 | \( 1 - 25501570 T^{2} + 286688776054767 T^{4} - \)\(21\!\cdots\!60\)\( T^{6} + 286688776054767 p^{10} T^{8} - 25501570 p^{20} T^{10} + p^{30} T^{12} \) |
| 29 | \( ( 1 - 11332 T + 103735252 T^{2} - 516534281986 T^{3} + 103735252 p^{5} T^{4} - 11332 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 31 | \( ( 1 - 9200 T + 105170221 T^{2} - 517958085024 T^{3} + 105170221 p^{5} T^{4} - 9200 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 37 | \( 1 + 1554670 T^{2} + 5217648302977047 T^{4} - \)\(31\!\cdots\!40\)\( T^{6} + 5217648302977047 p^{10} T^{8} + 1554670 p^{20} T^{10} + p^{30} T^{12} \) |
| 41 | \( ( 1 - 10442 T - 57000581 T^{2} + 56808109004 p T^{3} - 57000581 p^{5} T^{4} - 10442 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 43 | \( 1 + 106685174 T^{2} + 20458346845224807 T^{4} + \)\(57\!\cdots\!32\)\( T^{6} + 20458346845224807 p^{10} T^{8} + 106685174 p^{20} T^{10} + p^{30} T^{12} \) |
| 47 | \( 1 - 694810488 T^{2} + 273202876415610712 T^{4} - \)\(75\!\cdots\!74\)\( T^{6} + 273202876415610712 p^{10} T^{8} - 694810488 p^{20} T^{10} + p^{30} T^{12} \) |
| 53 | \( 1 - 1720738362 T^{2} + 1447541708221805287 T^{4} - \)\(75\!\cdots\!36\)\( T^{6} + 1447541708221805287 p^{10} T^{8} - 1720738362 p^{20} T^{10} + p^{30} T^{12} \) |
| 59 | \( ( 1 + 23760 T + 1782045217 T^{2} + 26972581939680 T^{3} + 1782045217 p^{5} T^{4} + 23760 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 61 | \( ( 1 - 8722 T + 2048748359 T^{2} - 16323927488596 T^{3} + 2048748359 p^{5} T^{4} - 8722 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 67 | \( 1 - 4459695618 T^{2} + 11130533234726608167 T^{4} - \)\(18\!\cdots\!04\)\( T^{6} + 11130533234726608167 p^{10} T^{8} - 4459695618 p^{20} T^{10} + p^{30} T^{12} \) |
| 71 | \( ( 1 + 52816 T + 4015281077 T^{2} + 113061643027552 T^{3} + 4015281077 p^{5} T^{4} + 52816 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 73 | \( 1 - 3503630762 T^{2} + 6075288961134011007 T^{4} - \)\(12\!\cdots\!16\)\( T^{6} + 6075288961134011007 p^{10} T^{8} - 3503630762 p^{20} T^{10} + p^{30} T^{12} \) |
| 79 | \( ( 1 - 71546 T + 8951915502 T^{2} - 399703392100308 T^{3} + 8951915502 p^{5} T^{4} - 71546 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 83 | \( 1 - 17802315762 T^{2} + \)\(14\!\cdots\!47\)\( T^{4} - \)\(73\!\cdots\!36\)\( T^{6} + \)\(14\!\cdots\!47\)\( p^{10} T^{8} - 17802315762 p^{20} T^{10} + p^{30} T^{12} \) |
| 89 | \( ( 1 - 192622 T + 24833946187 T^{2} - 2119564435111516 T^{3} + 24833946187 p^{5} T^{4} - 192622 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 97 | \( 1 - 23047495328 T^{2} + \)\(21\!\cdots\!32\)\( T^{4} - \)\(14\!\cdots\!34\)\( T^{6} + \)\(21\!\cdots\!32\)\( p^{10} T^{8} - 23047495328 p^{20} T^{10} + p^{30} T^{12} \) |
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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
−6.31859801354045903429659354148, −5.93547197521788488256462870172, −5.69580808011570059471860626178, −5.30262011707801652130071984930, −5.09551550208301949058260637962, −4.98981084175471849237755740573, −4.63607845840706983005983162373, −4.61177909877158855822123279234, −4.49734613946772855942770068313, −4.36324795419219386459260328387, −4.07011218558359365430216461546, −3.68962433670604174535477041049, −3.60502221383598277049318531549, −3.15925105116310508789901639238, −2.75816919289504262955761871232, −2.71152204756650255015974097108, −2.40452949814561213528104462474, −2.33853344502133918074769726937, −2.31180405931527734557259404806, −1.33761294040630824192441458087, −1.17441413763981119662787363279, −1.15637923402801320822718100008, −1.00981244982080364852692208448, −0.37146906104052788829890639299, −0.14272320105726429515297437636,
0.14272320105726429515297437636, 0.37146906104052788829890639299, 1.00981244982080364852692208448, 1.15637923402801320822718100008, 1.17441413763981119662787363279, 1.33761294040630824192441458087, 2.31180405931527734557259404806, 2.33853344502133918074769726937, 2.40452949814561213528104462474, 2.71152204756650255015974097108, 2.75816919289504262955761871232, 3.15925105116310508789901639238, 3.60502221383598277049318531549, 3.68962433670604174535477041049, 4.07011218558359365430216461546, 4.36324795419219386459260328387, 4.49734613946772855942770068313, 4.61177909877158855822123279234, 4.63607845840706983005983162373, 4.98981084175471849237755740573, 5.09551550208301949058260637962, 5.30262011707801652130071984930, 5.69580808011570059471860626178, 5.93547197521788488256462870172, 6.31859801354045903429659354148
Plot not available for L-functions of degree greater than 10.
