| L(s) = 1 | + 2·4-s − 4·9-s + 8·11-s + 16-s + 4·19-s − 25-s − 8·31-s − 8·36-s − 4·41-s + 16·44-s − 4·49-s + 24·59-s − 8·61-s − 2·64-s + 44·71-s + 8·76-s − 4·79-s + 22·81-s + 20·89-s − 32·99-s − 2·100-s − 96·101-s − 24·109-s − 8·121-s − 16·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 4-s − 4/3·9-s + 2.41·11-s + 1/4·16-s + 0.917·19-s − 1/5·25-s − 1.43·31-s − 4/3·36-s − 0.624·41-s + 2.41·44-s − 4/7·49-s + 3.12·59-s − 1.02·61-s − 1/4·64-s + 5.22·71-s + 0.917·76-s − 0.450·79-s + 22/9·81-s + 2.11·89-s − 3.21·99-s − 1/5·100-s − 9.55·101-s − 2.29·109-s − 0.727·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.312374252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.312374252\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 5 | \( 1 + T^{2} - 24 T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | \( ( 1 + 47 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 + 4 T^{2} - 6 p T^{4} - 160 T^{6} + 179 T^{8} - 160 p^{2} T^{10} - 6 p^{5} T^{12} + 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | \( 1 + 38 T^{2} + 681 T^{4} + 7030 T^{6} + 81332 T^{8} + 7030 p^{2} T^{10} + 681 p^{4} T^{12} + 38 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 2 T - 24 T^{2} + 20 T^{3} + 347 T^{4} + 20 p T^{5} - 24 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 47 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 2 T + 52 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 2 T - 35 T^{2} - 86 T^{3} - 324 T^{4} - 86 p T^{5} - 35 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 28 T^{2} + 3498 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 164 T^{2} + 11098 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 + 116 T^{2} + 5178 T^{4} + 308560 T^{6} + 22836899 T^{8} + 308560 p^{2} T^{10} + 5178 p^{4} T^{12} + 116 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 12 T + 34 T^{2} + 96 T^{3} + 123 T^{4} + 96 p T^{5} + 34 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 4 T - 11 T^{2} - 380 T^{3} - 3968 T^{4} - 380 p T^{5} - 11 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 196 T^{2} + 20934 T^{4} + 1666784 T^{6} + 113019779 T^{8} + 1666784 p^{2} T^{10} + 20934 p^{4} T^{12} + 196 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 22 T + 232 T^{2} - 2420 T^{3} + 24099 T^{4} - 2420 p T^{5} + 232 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 172 T^{2} + 15238 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 2 T - 56 T^{2} - 196 T^{3} - 2957 T^{4} - 196 p T^{5} - 56 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 122 T^{2} + 7995 T^{4} + 122 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 10 T - 59 T^{2} + 190 T^{3} + 8460 T^{4} + 190 p T^{5} - 59 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 17 T + p T^{2} )^{4}( 1 + 17 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.19313195971648161798496513629, −4.96860297197689377159560756307, −4.89849127946528970334212524799, −4.60623432720673260663510940503, −4.51497283382260931273053777047, −4.20150589882799511009921603555, −4.14559257955724824228646713403, −3.90661864910432203478654490584, −3.84610977660767159853472137906, −3.61636346636067079982503763048, −3.58839257132454997498050742682, −3.45535367933090608461135603418, −3.33565392819426832011773052291, −3.11902955785844298031534569268, −2.62840298641394066005681522551, −2.58039864961490559864168420799, −2.52054931702342127146291312207, −2.49877187507108648673795629662, −2.01058765060270737050008591613, −1.97308957337219426172340879660, −1.66496182577626167655062952994, −1.22325277491720704754375292076, −1.21559847215003377504620202141, −1.06230187051445519253862898191, −0.29182862757285754836733486793,
0.29182862757285754836733486793, 1.06230187051445519253862898191, 1.21559847215003377504620202141, 1.22325277491720704754375292076, 1.66496182577626167655062952994, 1.97308957337219426172340879660, 2.01058765060270737050008591613, 2.49877187507108648673795629662, 2.52054931702342127146291312207, 2.58039864961490559864168420799, 2.62840298641394066005681522551, 3.11902955785844298031534569268, 3.33565392819426832011773052291, 3.45535367933090608461135603418, 3.58839257132454997498050742682, 3.61636346636067079982503763048, 3.84610977660767159853472137906, 3.90661864910432203478654490584, 4.14559257955724824228646713403, 4.20150589882799511009921603555, 4.51497283382260931273053777047, 4.60623432720673260663510940503, 4.89849127946528970334212524799, 4.96860297197689377159560756307, 5.19313195971648161798496513629
Plot not available for L-functions of degree greater than 10.
