| L(s) = 1 | − 16·13-s − 16·25-s − 16·37-s − 24·49-s − 48·61-s + 16·73-s − 32·97-s − 48·109-s − 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 48·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 4.43·13-s − 3.19·25-s − 2.63·37-s − 3.42·49-s − 6.14·61-s + 1.87·73-s − 3.24·97-s − 4.59·109-s − 5.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 8 T^{2} + 58 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + 12 T^{2} + 18 p T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 28 T^{2} + 406 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 28 T^{2} + 486 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 12 T^{2} - 42 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 60 T^{2} + 1830 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 40 T^{2} + 2010 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 12 T^{2} + 1566 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4 T + 76 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 60 T^{2} + 1670 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 44 T^{2} + 310 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 28 T^{2} - 1658 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 136 T^{2} + 10170 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 108 T^{2} + 7830 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 12 T + 156 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 140 T^{2} + 13366 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 28 T^{2} + 2086 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 300 T^{2} + 34974 T^{4} + 300 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 60 T^{2} + 13110 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 164 T^{2} + 20518 T^{4} + 164 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.62945649253599639324324259610, −3.30912662283415262689290275528, −3.20425328997762103991527135088, −3.12773597031521965065441796519, −3.11520326053564171638371824459, −3.11050851432847909949671603660, −2.90883470652826791133515208242, −2.75673595033815677415665395277, −2.53922421101828324335147852573, −2.50146677701074022771787973263, −2.48661893085262783878053563364, −2.36741131406020126598444634292, −2.29532877446298614768111528992, −2.17416241288420794410422987901, −2.07204178580238203307874869991, −2.03826695715910446825070625950, −1.80132477552453758925093022613, −1.71745039928855341021262487554, −1.46401738830977127382449277146, −1.45967594842155827029236497325, −1.27982181704772264413541696776, −1.19526135921505401671685618154, −1.15712006402015704986724132650, −1.04199181547917274150240206531, −0.913489669257919157726584783785, 0, 0, 0, 0, 0, 0, 0, 0,
0.913489669257919157726584783785, 1.04199181547917274150240206531, 1.15712006402015704986724132650, 1.19526135921505401671685618154, 1.27982181704772264413541696776, 1.45967594842155827029236497325, 1.46401738830977127382449277146, 1.71745039928855341021262487554, 1.80132477552453758925093022613, 2.03826695715910446825070625950, 2.07204178580238203307874869991, 2.17416241288420794410422987901, 2.29532877446298614768111528992, 2.36741131406020126598444634292, 2.48661893085262783878053563364, 2.50146677701074022771787973263, 2.53922421101828324335147852573, 2.75673595033815677415665395277, 2.90883470652826791133515208242, 3.11050851432847909949671603660, 3.11520326053564171638371824459, 3.12773597031521965065441796519, 3.20425328997762103991527135088, 3.30912662283415262689290275528, 3.62945649253599639324324259610
Plot not available for L-functions of degree greater than 10.
