| L(s) = 1 | − 4·25-s − 32·41-s + 48·49-s − 36·81-s − 16·89-s + 72·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 24·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 4/5·25-s − 4.99·41-s + 48/7·49-s − 4·81-s − 1.69·89-s + 6.54·121-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 + 1.84·169-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 + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.350790496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.350790496\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 43 | \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 76 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + p T^{2} )^{8} \) |
| 59 | \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 97 | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−5.22897475258815021567475848221, −5.05116399024497411157962406659, −4.96603103366504215898698420302, −4.86721350314232118735051269196, −4.44542507118119795734508654157, −4.40293352949898722162853413851, −4.37341784841127647492458158909, −4.18454433343679117702411069805, −3.90293463602792371090500467811, −3.85017615613023600042233018883, −3.81272608323679950141687032036, −3.41959735746845665314996242588, −3.28437876669398359447693629076, −3.21517462654926945323503370272, −2.97706620355683093438152571263, −2.80572461435189366092514981619, −2.52952004466863090983915260680, −2.40054514738793687255700269314, −2.19495323186898686631424848099, −1.91140774477372873124728056901, −1.74640248866218297015733100107, −1.46150173448109023417904371919, −1.33870051875073351926471888572, −0.65906081523396271103566142236, −0.48944471103608752263182176230,
0.48944471103608752263182176230, 0.65906081523396271103566142236, 1.33870051875073351926471888572, 1.46150173448109023417904371919, 1.74640248866218297015733100107, 1.91140774477372873124728056901, 2.19495323186898686631424848099, 2.40054514738793687255700269314, 2.52952004466863090983915260680, 2.80572461435189366092514981619, 2.97706620355683093438152571263, 3.21517462654926945323503370272, 3.28437876669398359447693629076, 3.41959735746845665314996242588, 3.81272608323679950141687032036, 3.85017615613023600042233018883, 3.90293463602792371090500467811, 4.18454433343679117702411069805, 4.37341784841127647492458158909, 4.40293352949898722162853413851, 4.44542507118119795734508654157, 4.86721350314232118735051269196, 4.96603103366504215898698420302, 5.05116399024497411157962406659, 5.22897475258815021567475848221
Plot not available for L-functions of degree greater than 10.
