| L(s) = 1 | − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s − 256-s + ⋯ |
| L(s) = 1 | − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s − 256-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.215112574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.215112574\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{8} \) |
| 3 | \( 1 \) |
| 37 | \( ( 1 + T^{2} )^{4} \) |
| good | 5 | \( ( 1 + T^{8} )^{2} \) |
| 7 | \( ( 1 + T^{4} )^{4} \) |
| 11 | \( ( 1 + T^{2} )^{8} \) |
| 13 | \( ( 1 + T^{2} )^{8} \) |
| 17 | \( ( 1 + T^{8} )^{2} \) |
| 19 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 23 | \( ( 1 + T^{8} )^{2} \) |
| 29 | \( ( 1 + T^{8} )^{2} \) |
| 31 | \( ( 1 + T^{2} )^{8} \) |
| 41 | \( ( 1 + T^{2} )^{8} \) |
| 43 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 47 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 53 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 59 | \( ( 1 + T^{8} )^{2} \) |
| 61 | \( ( 1 + T^{2} )^{8} \) |
| 67 | \( ( 1 + T^{4} )^{4} \) |
| 71 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 73 | \( ( 1 + T^{4} )^{4} \) |
| 79 | \( ( 1 + T^{2} )^{8} \) |
| 83 | \( ( 1 + T^{2} )^{8} \) |
| 89 | \( ( 1 + T^{8} )^{2} \) |
| 97 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
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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
−3.80449999953140898294236880236, −3.79818255645462344592512164493, −3.73357428342909657581953319407, −3.72433079740208203324357275339, −3.64259185657015584920750699319, −3.24879307978255272710451852445, −3.04286150509872071030676768868, −3.03086757903055465057292838216, −2.95350266444204681305656460230, −2.79087697507718456370747144481, −2.59515814488071465684412498713, −2.59298913796615371330345718112, −2.58386153475641813255657178739, −2.50803906633561769262119427533, −2.16275827221005402632779570717, −1.97360262838182725643267227920, −1.79208364576888426890910647434, −1.60554284171153691975487844408, −1.59495025831135935076569858307, −1.55610531745039949362642044634, −1.48576644745438471362919550044, −0.959552126782797026366002181030, −0.874008472681171131240413780436, −0.795240026356658582282098779273, −0.35051348728392832652617393472,
0.35051348728392832652617393472, 0.795240026356658582282098779273, 0.874008472681171131240413780436, 0.959552126782797026366002181030, 1.48576644745438471362919550044, 1.55610531745039949362642044634, 1.59495025831135935076569858307, 1.60554284171153691975487844408, 1.79208364576888426890910647434, 1.97360262838182725643267227920, 2.16275827221005402632779570717, 2.50803906633561769262119427533, 2.58386153475641813255657178739, 2.59298913796615371330345718112, 2.59515814488071465684412498713, 2.79087697507718456370747144481, 2.95350266444204681305656460230, 3.03086757903055465057292838216, 3.04286150509872071030676768868, 3.24879307978255272710451852445, 3.64259185657015584920750699319, 3.72433079740208203324357275339, 3.73357428342909657581953319407, 3.79818255645462344592512164493, 3.80449999953140898294236880236
Plot not available for L-functions of degree greater than 10.
