| L(s) = 1 | + 12·9-s + 32·11-s + 16·17-s + 96·19-s + 104·25-s − 80·41-s + 224·43-s + 152·49-s − 512·59-s + 16·73-s + 90·81-s − 544·83-s + 240·89-s + 400·97-s + 384·99-s − 448·107-s + 336·113-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 192·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 4/3·9-s + 2.90·11-s + 0.941·17-s + 5.05·19-s + 4.15·25-s − 1.95·41-s + 5.20·43-s + 3.10·49-s − 8.67·59-s + 0.219·73-s + 10/9·81-s − 6.55·83-s + 2.69·89-s + 4.12·97-s + 3.87·99-s − 4.18·107-s + 2.97·113-s − 0.0661·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 1.25·153-s + 0.00636·157-s + 0.00613·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(8.942831673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.942831673\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - p T^{2} )^{4} \) |
| good | 5 | \( 1 - 104 T^{2} + 5148 T^{4} - 167512 T^{6} + 4425734 T^{8} - 167512 p^{4} T^{10} + 5148 p^{8} T^{12} - 104 p^{12} T^{14} + p^{16} T^{16} \) |
| 7 | \( 1 - 152 T^{2} + 1476 p T^{4} - 514600 T^{6} + 25187654 T^{8} - 514600 p^{4} T^{10} + 1476 p^{9} T^{12} - 152 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( ( 1 - 8 T + 162 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 13 | \( 1 - 504 T^{2} + 15052 p T^{4} - 48291912 T^{6} + 9653428230 T^{8} - 48291912 p^{4} T^{10} + 15052 p^{9} T^{12} - 504 p^{12} T^{14} + p^{16} T^{16} \) |
| 17 | \( ( 1 - 8 T + 252 T^{2} + 2888 T^{3} + 48902 T^{4} + 2888 p^{2} T^{5} + 252 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 48 T + 1764 T^{2} - 39696 T^{3} + 863462 T^{4} - 39696 p^{2} T^{5} + 1764 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 - 3272 T^{2} + 5007132 T^{4} - 4715977336 T^{6} + 5669682278 p^{2} T^{8} - 4715977336 p^{4} T^{10} + 5007132 p^{8} T^{12} - 3272 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 - 2792 T^{2} + 5333148 T^{4} - 6634744024 T^{6} + 6540835240070 T^{8} - 6634744024 p^{4} T^{10} + 5333148 p^{8} T^{12} - 2792 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( 1 - 4248 T^{2} + 9652444 T^{4} - 14917063464 T^{6} + 16677690731718 T^{8} - 14917063464 p^{4} T^{10} + 9652444 p^{8} T^{12} - 4248 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( 1 - 3896 T^{2} + 7037532 T^{4} - 10040494600 T^{6} + 14385716443526 T^{8} - 10040494600 p^{4} T^{10} + 7037532 p^{8} T^{12} - 3896 p^{12} T^{14} + p^{16} T^{16} \) |
| 41 | \( ( 1 + 40 T + 2556 T^{2} - 43240 T^{3} - 289594 T^{4} - 43240 p^{2} T^{5} + 2556 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 112 T + 9252 T^{2} - 494288 T^{3} + 24081062 T^{4} - 494288 p^{2} T^{5} + 9252 p^{4} T^{6} - 112 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 - 12360 T^{2} + 70380700 T^{4} - 250403346936 T^{6} + 638353123484742 T^{8} - 250403346936 p^{4} T^{10} + 70380700 p^{8} T^{12} - 12360 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( 1 - 12392 T^{2} + 85017372 T^{4} - 388365545560 T^{6} + 1274899611031814 T^{8} - 388365545560 p^{4} T^{10} + 85017372 p^{8} T^{12} - 12392 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( ( 1 + 256 T + 36612 T^{2} + 3480320 T^{3} + 239708390 T^{4} + 3480320 p^{2} T^{5} + 36612 p^{4} T^{6} + 256 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 61 | \( 1 - 9656 T^{2} + 53408988 T^{4} - 170291139208 T^{6} + 575235456654854 T^{8} - 170291139208 p^{4} T^{10} + 53408988 p^{8} T^{12} - 9656 p^{12} T^{14} + p^{16} T^{16} \) |
| 67 | \( ( 1 + 8644 T^{2} + 58097190 T^{4} + 8644 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( 1 - 18120 T^{2} + 152088604 T^{4} - 903092412024 T^{6} + 4737261672397254 T^{8} - 903092412024 p^{4} T^{10} + 152088604 p^{8} T^{12} - 18120 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 - 8 T + 5212 T^{2} - 653240 T^{3} - 3523322 T^{4} - 653240 p^{2} T^{5} + 5212 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( 1 - 26776 T^{2} + 333200860 T^{4} - 2815930225960 T^{6} + 19175645821896646 T^{8} - 2815930225960 p^{4} T^{10} + 333200860 p^{8} T^{12} - 26776 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( ( 1 + 272 T + 44868 T^{2} + 5460400 T^{3} + 513991334 T^{4} + 5460400 p^{2} T^{5} + 44868 p^{4} T^{6} + 272 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 120 T + 30300 T^{2} - 2699976 T^{3} + 352873862 T^{4} - 2699976 p^{2} T^{5} + 30300 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 200 T + 21276 T^{2} - 3009400 T^{3} + 385130822 T^{4} - 3009400 p^{2} T^{5} + 21276 p^{4} T^{6} - 200 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−4.23986927274082502862334038211, −4.23485905252405623703116380208, −3.86356033461693924928385872153, −3.78545747866372587537621842660, −3.69239314990055361126645869615, −3.53474641938904486264938306992, −3.42858649887955283300868866566, −3.25041601879863790242149026798, −3.01668856519004661866681076881, −2.97452503028496247576999432677, −2.91817761448426790363007188672, −2.83121776488809841381321373329, −2.58500709697493714937025706959, −2.51153292759672768766481950477, −2.08072851930310967450084587219, −1.93283984670340245992777814863, −1.65485461872117369340464436157, −1.32409520216906146461334118653, −1.31720028970895984879773677167, −1.30821093558279387055074714618, −1.19421680390314747396562095121, −0.965574797088064744442178046107, −0.77825587207793139429770356020, −0.77277943973742958823157742571, −0.12329388684400947830568492191,
0.12329388684400947830568492191, 0.77277943973742958823157742571, 0.77825587207793139429770356020, 0.965574797088064744442178046107, 1.19421680390314747396562095121, 1.30821093558279387055074714618, 1.31720028970895984879773677167, 1.32409520216906146461334118653, 1.65485461872117369340464436157, 1.93283984670340245992777814863, 2.08072851930310967450084587219, 2.51153292759672768766481950477, 2.58500709697493714937025706959, 2.83121776488809841381321373329, 2.91817761448426790363007188672, 2.97452503028496247576999432677, 3.01668856519004661866681076881, 3.25041601879863790242149026798, 3.42858649887955283300868866566, 3.53474641938904486264938306992, 3.69239314990055361126645869615, 3.78545747866372587537621842660, 3.86356033461693924928385872153, 4.23485905252405623703116380208, 4.23986927274082502862334038211
Plot not available for L-functions of degree greater than 10.
