| L(s) = 1 | + 16·17-s + 40·25-s − 144·41-s + 360·49-s − 80·73-s + 272·89-s − 528·97-s − 656·113-s − 264·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 904·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 0.941·17-s + 8/5·25-s − 3.51·41-s + 7.34·49-s − 1.09·73-s + 3.05·89-s − 5.44·97-s − 5.80·113-s − 2.18·121-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 + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 5.34·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7033225642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7033225642\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 4 p T^{2} - 186 T^{4} - 4 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 - 90 T^{2} + p^{4} T^{4} )^{4} \) |
| 11 | \( ( 1 + 12 p T^{2} + 9062 T^{4} + 12 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 452 T^{2} + 102054 T^{4} - 452 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 4 T + 198 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 + 1092 T^{2} + 534182 T^{4} + 1092 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 516 T^{2} + 998 p^{2} T^{4} - 516 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 1364 T^{2} + 919686 T^{4} - 1364 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 1012 T^{2} + 112422 T^{4} - 1012 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2372 T^{2} + 5056614 T^{4} - 2372 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 36 T + 3302 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 + 900 T^{2} + 6425702 T^{4} + 900 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 2116 T^{2} + 7339782 T^{4} - 2116 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 6292 T^{2} + 19581894 T^{4} - 6292 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 5762 T^{2} + p^{4} T^{4} )^{4} \) |
| 61 | \( ( 1 - 10244 T^{2} + 50952870 T^{4} - 10244 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 9156 T^{2} + 41992742 T^{4} + 9156 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 132 T^{2} + 22417862 T^{4} - 132 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 + 10 T + p^{2} T^{2} )^{8} \) |
| 79 | \( ( 1 - 6516 T^{2} + 24592550 T^{4} - 6516 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 + 13636 T^{2} + 114638502 T^{4} + 13636 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 68 T + 15462 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 + 132 T + 21638 T^{2} + 132 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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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
−4.02508327626081786694368661513, −3.66252682945640314197790963145, −3.64946690769794039457785569738, −3.64118908673327097126789974055, −3.47302918085435419570637535398, −3.44028561851027006988067331774, −3.14547347215066988069520483740, −2.95207141629920803379418016012, −2.79433873926281439210583594860, −2.79130693577864188869297414258, −2.59992719157262502180371709321, −2.55560909186691030043161322201, −2.29078020538055217567660252048, −2.18569705934990931900120175901, −2.10078972520905259451901422199, −1.87865466543705960701325088973, −1.63932116951931649320045668130, −1.36625606762420919497736233700, −1.24023446148577951777020393494, −1.07952227682514234273779746952, −1.05346163124280722594841766748, −0.972867702153550166053926435609, −0.60859346707266048397303410690, −0.16678798527854457652783393846, −0.11077248205690971206278527659,
0.11077248205690971206278527659, 0.16678798527854457652783393846, 0.60859346707266048397303410690, 0.972867702153550166053926435609, 1.05346163124280722594841766748, 1.07952227682514234273779746952, 1.24023446148577951777020393494, 1.36625606762420919497736233700, 1.63932116951931649320045668130, 1.87865466543705960701325088973, 2.10078972520905259451901422199, 2.18569705934990931900120175901, 2.29078020538055217567660252048, 2.55560909186691030043161322201, 2.59992719157262502180371709321, 2.79130693577864188869297414258, 2.79433873926281439210583594860, 2.95207141629920803379418016012, 3.14547347215066988069520483740, 3.44028561851027006988067331774, 3.47302918085435419570637535398, 3.64118908673327097126789974055, 3.64946690769794039457785569738, 3.66252682945640314197790963145, 4.02508327626081786694368661513
Plot not available for L-functions of degree greater than 10.
