| L(s) = 1 | + 4·3-s + 8·9-s + 96·13-s + 16·19-s − 12·27-s − 72·31-s − 112·37-s + 384·39-s − 240·43-s + 360·49-s + 64·57-s − 208·61-s − 232·67-s + 136·79-s − 34·81-s − 288·93-s − 480·97-s − 64·109-s − 448·111-s + 768·117-s + 127-s − 960·129-s + 131-s + 137-s + 139-s + 1.44e3·147-s + 149-s + ⋯ |
| L(s) = 1 | + 4/3·3-s + 8/9·9-s + 7.38·13-s + 0.842·19-s − 4/9·27-s − 2.32·31-s − 3.02·37-s + 9.84·39-s − 5.58·43-s + 7.34·49-s + 1.12·57-s − 3.40·61-s − 3.46·67-s + 1.72·79-s − 0.419·81-s − 3.09·93-s − 4.94·97-s − 0.587·109-s − 4.03·111-s + 6.56·117-s + 0.00787·127-s − 7.44·129-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 9.79·147-s + 0.00671·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(13.32205523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.32205523\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 4 T + 8 T^{2} + 4 p T^{3} - 14 p^{2} T^{4} + 4 p^{3} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| good | 5 | \( 1 + 372 T^{4} + 464614 T^{8} + 372 p^{8} T^{12} + p^{16} T^{16} \) |
| 7 | \( ( 1 - 180 T^{2} + 12874 T^{4} - 180 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( 1 + 37380 T^{4} + 692870854 T^{8} + 37380 p^{8} T^{12} + p^{16} T^{16} \) |
| 13 | \( ( 1 - 48 T + 1152 T^{2} - 21264 T^{3} + 317422 T^{4} - 21264 p^{2} T^{5} + 1152 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 236 T^{2} + 73334 T^{4} - 236 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 8 T + 32 T^{2} - 1160 T^{3} - 4606 T^{4} - 1160 p^{2} T^{5} + 32 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 + 996 T^{2} + 719878 T^{4} + 996 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( 1 + 147060 T^{4} + 623812106854 T^{8} + 147060 p^{8} T^{12} + p^{16} T^{16} \) |
| 31 | \( ( 1 + 18 T + 1996 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 + 56 T + 1568 T^{2} + 79016 T^{3} + 3980078 T^{4} + 79016 p^{2} T^{5} + 1568 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 6060 T^{2} + 14724790 T^{4} + 6060 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 120 T + 7200 T^{2} + 411000 T^{3} + 20977474 T^{4} + 411000 p^{2} T^{5} + 7200 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 692 T^{2} + 7491686 T^{4} - 692 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( 1 + 1441524 T^{4} - 91064986479386 T^{8} + 1441524 p^{8} T^{12} + p^{16} T^{16} \) |
| 59 | \( 1 + 15787044 T^{4} + 346992256453126 T^{8} + 15787044 p^{8} T^{12} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 104 T + 5408 T^{2} + 491192 T^{3} + 43609454 T^{4} + 491192 p^{2} T^{5} + 5408 p^{4} T^{6} + 104 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 116 T + 6728 T^{2} + 350436 T^{3} + 16097858 T^{4} + 350436 p^{2} T^{5} + 6728 p^{4} T^{6} + 116 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 + 3604 T^{2} + 10180454 T^{4} + 3604 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 17604 T^{2} + 131615494 T^{4} - 17604 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 34 T + 11588 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 83 | \( 1 - 64624380 T^{4} + 2364945173919814 T^{8} - 64624380 p^{8} T^{12} + p^{16} T^{16} \) |
| 89 | \( ( 1 + 25444 T^{2} + 277784198 T^{4} + 25444 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 120 T + 21970 T^{2} + 120 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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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
−4.63231053605361000194557309691, −4.47613454595352772615037930208, −4.45180634594873499385390140718, −4.16815163460470363243155043612, −4.04881581242913440497028024931, −4.03489805672819057923152171906, −3.69896975915131004828520073306, −3.59826512079401137226247024826, −3.59223443223298637972701664865, −3.41968031416499195340733035861, −3.40896074562647762205375518995, −3.15409435807700063500726740430, −3.00463128136176705749042828518, −2.93608131419209932126482632523, −2.71085451786331077872787149087, −2.24952977466018928387540103187, −1.86535848049680853297636676708, −1.85049573281990042813794351540, −1.68346271949407370012550905130, −1.67178154756992308866850399460, −1.38516142138904046657045363003, −1.27526892548279821762718348561, −0.820575488530630287152921197145, −0.74334028281767689686945140282, −0.23824842545364338831095578937,
0.23824842545364338831095578937, 0.74334028281767689686945140282, 0.820575488530630287152921197145, 1.27526892548279821762718348561, 1.38516142138904046657045363003, 1.67178154756992308866850399460, 1.68346271949407370012550905130, 1.85049573281990042813794351540, 1.86535848049680853297636676708, 2.24952977466018928387540103187, 2.71085451786331077872787149087, 2.93608131419209932126482632523, 3.00463128136176705749042828518, 3.15409435807700063500726740430, 3.40896074562647762205375518995, 3.41968031416499195340733035861, 3.59223443223298637972701664865, 3.59826512079401137226247024826, 3.69896975915131004828520073306, 4.03489805672819057923152171906, 4.04881581242913440497028024931, 4.16815163460470363243155043612, 4.45180634594873499385390140718, 4.47613454595352772615037930208, 4.63231053605361000194557309691
Plot not available for L-functions of degree greater than 10.
