| L(s) = 1 | − 24·9-s − 32·17-s − 4·25-s + 20·49-s − 96·73-s + 324·81-s − 32·97-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 768·153-s + 157-s + 163-s + 167-s + 56·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 8·9-s − 7.76·17-s − 4/5·25-s + 20/7·49-s − 11.2·73-s + 36·81-s − 3.24·97-s − 3.63·121-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 + 62.0·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4.30·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04245712977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04245712977\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 + p T^{2} )^{8} \) |
| 11 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 19 | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 126 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 12 T + p T^{2} )^{8} \) |
| 79 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
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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
−3.87371142438470900871387893920, −3.81085790908845802243471281131, −3.46478396092198670178042549865, −3.22508472548404495056126184831, −3.12064631059228862798623982358, −2.99338588696692329861982467335, −2.89923982045270333190662049790, −2.84346308426753792653516428143, −2.83900122292906336717668199125, −2.62898746711439258320554918303, −2.62028404681448552757743399701, −2.48577086849726888549779766039, −2.37079370679177016053650770770, −2.36406552945223601435610157316, −1.96934883069273378117570973959, −1.89610849093082700472594525252, −1.84339255998328688236901968066, −1.77609028711002670821622013689, −1.42873940100493837803346371059, −1.24043430860096759978655997122, −0.71746611807553145233072609718, −0.47316010008235558441415532362, −0.35645645953996597267013078947, −0.22729809404443584290115306947, −0.10391773069478189563154307978,
0.10391773069478189563154307978, 0.22729809404443584290115306947, 0.35645645953996597267013078947, 0.47316010008235558441415532362, 0.71746611807553145233072609718, 1.24043430860096759978655997122, 1.42873940100493837803346371059, 1.77609028711002670821622013689, 1.84339255998328688236901968066, 1.89610849093082700472594525252, 1.96934883069273378117570973959, 2.36406552945223601435610157316, 2.37079370679177016053650770770, 2.48577086849726888549779766039, 2.62028404681448552757743399701, 2.62898746711439258320554918303, 2.83900122292906336717668199125, 2.84346308426753792653516428143, 2.89923982045270333190662049790, 2.99338588696692329861982467335, 3.12064631059228862798623982358, 3.22508472548404495056126184831, 3.46478396092198670178042549865, 3.81085790908845802243471281131, 3.87371142438470900871387893920
Plot not available for L-functions of degree greater than 10.
