| L(s) = 1 | + 8·13-s − 4·25-s + 48·37-s − 20·49-s + 8·61-s − 48·73-s − 40·97-s − 144·109-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 76·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 2.21·13-s − 4/5·25-s + 7.89·37-s − 2.85·49-s + 1.02·61-s − 5.61·73-s − 4.06·97-s − 13.7·109-s + 2.54·121-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 + 5.84·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.551042579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.551042579\) |
| \(L(\frac{3}{2})\) |
|
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 | \( ( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 14 T^{2} + 75 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 7 T + p T^{2} )^{4}( 1 + 5 T + p T^{2} )^{4} \) |
| 17 | \( ( 1 - p T^{2} )^{8} \) |
| 19 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 14 T^{2} - 333 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 50 T^{2} + 1659 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 58 T^{2} + 2403 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 6 T + p T^{2} )^{8} \) |
| 41 | \( ( 1 + 50 T^{2} + 819 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 82 T^{2} + 4875 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 34 T^{2} - 1053 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 110 T^{2} + 8619 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 130 T^{2} + 12411 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 6 T + p T^{2} )^{8} \) |
| 79 | \( ( 1 - 38 T^{2} - 4797 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 158 T^{2} + 18075 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.81186167162098117519020802128, −3.75453584965584673818776003744, −3.47631789447593650356972446553, −3.13367173471261597716914016206, −3.12124811346351574152347038155, −3.09912574663871891200449841805, −2.88747660692505604855823144083, −2.77255645348315540028526384215, −2.70637039956417355855745656734, −2.70381502295607878956433401016, −2.65811166387654362206194989230, −2.56815073879594872251932408948, −2.20181157120466646990534716546, −1.93850323544206255273071671072, −1.85224502674862624640689973841, −1.73199582666442101665696887790, −1.67343580243633774692765711856, −1.30894431912273558548740563253, −1.27500338240728386290810802776, −1.25136627720119561209121655697, −1.01669137576574921274726427430, −1.00672954437088836917923906201, −0.58308446437201485930556830070, −0.33897340861268764360857477232, −0.19407022268631289996401194037,
0.19407022268631289996401194037, 0.33897340861268764360857477232, 0.58308446437201485930556830070, 1.00672954437088836917923906201, 1.01669137576574921274726427430, 1.25136627720119561209121655697, 1.27500338240728386290810802776, 1.30894431912273558548740563253, 1.67343580243633774692765711856, 1.73199582666442101665696887790, 1.85224502674862624640689973841, 1.93850323544206255273071671072, 2.20181157120466646990534716546, 2.56815073879594872251932408948, 2.65811166387654362206194989230, 2.70381502295607878956433401016, 2.70637039956417355855745656734, 2.77255645348315540028526384215, 2.88747660692505604855823144083, 3.09912574663871891200449841805, 3.12124811346351574152347038155, 3.13367173471261597716914016206, 3.47631789447593650356972446553, 3.75453584965584673818776003744, 3.81186167162098117519020802128
Plot not available for L-functions of degree greater than 10.
