| L(s) = 1 | − 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 + 544·83-s − 240·89-s + 400·97-s + 448·107-s − 336·113-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 504·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 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 + 6.55·83-s − 2.69·89-s + 4.12·97-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 + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.98·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(96.01201242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(96.01201242\) |
| \(L(2)\) |
|
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 | \( 1 - 104 T^{2} + 5148 T^{4} - 167512 T^{6} + 4425734 T^{8} - 167512 p^{4} T^{10} + 5148 p^{8} T^{12} - 104 p^{12} T^{14} + p^{16} T^{16} \) |
| 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
−3.56973385499537453755880207284, −3.16799847486269374581430298621, −3.16182692626780785604144969334, −3.12464452205437913839372345378, −3.07492190605509947167884681561, −2.98149981119600202269919023448, −2.94982293268935034374808489536, −2.63135997253282899583759804919, −2.59208060427383449432378426475, −2.41376806077273895844683150620, −2.34617392995307042307688358377, −2.17564437960394385498553399956, −2.15571437042288559105393110833, −2.11226945474570591394578935607, −1.90731114799011778675478976787, −1.62537491323022026127880302280, −1.31791274816234427573884764693, −0.949242762933844009594424728187, −0.947268273921879920414276990202, −0.913245436255730941762757592894, −0.797773314815004366025184356609, −0.73100815080844850716677567365, −0.69269790404052235843934253777, −0.53905655620381741579346125094, −0.34943590090934267494607594539,
0.34943590090934267494607594539, 0.53905655620381741579346125094, 0.69269790404052235843934253777, 0.73100815080844850716677567365, 0.797773314815004366025184356609, 0.913245436255730941762757592894, 0.947268273921879920414276990202, 0.949242762933844009594424728187, 1.31791274816234427573884764693, 1.62537491323022026127880302280, 1.90731114799011778675478976787, 2.11226945474570591394578935607, 2.15571437042288559105393110833, 2.17564437960394385498553399956, 2.34617392995307042307688358377, 2.41376806077273895844683150620, 2.59208060427383449432378426475, 2.63135997253282899583759804919, 2.94982293268935034374808489536, 2.98149981119600202269919023448, 3.07492190605509947167884681561, 3.12464452205437913839372345378, 3.16182692626780785604144969334, 3.16799847486269374581430298621, 3.56973385499537453755880207284
Plot not available for L-functions of degree greater than 10.
