| L(s) = 1 | + 8·4-s + 24·16-s + 72·23-s − 28·49-s + 20·81-s + 576·92-s + 88·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s − 224·196-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 4·4-s + 6·16-s + 15.0·23-s − 4·49-s + 20/9·81-s + 60.0·92-s + 8·121-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 16·196-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(31.39987930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(31.39987930\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 3 | \( ( 1 - 10 T^{4} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| good | 5 | \( ( 1 + 22 T^{4} + p^{4} T^{8} )^{2}( 1 - 22 T^{4} - 141 T^{8} - 22 p^{4} T^{12} + p^{8} T^{16} ) \) |
| 11 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{8} \) |
| 13 | \( ( 1 - 310 T^{4} + 67539 T^{8} - 310 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 + p T^{2} )^{16} \) |
| 19 | \( ( 1 + 650 T^{4} + 292179 T^{8} + 650 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 - 6 T + p T^{2} )^{8}( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{8} \) |
| 31 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{8} \) |
| 37 | \( ( 1 - p T^{2} )^{16} \) |
| 41 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{8} \) |
| 43 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{8} \) |
| 47 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{8} \) |
| 53 | \( ( 1 + p T^{2} )^{16} \) |
| 59 | \( ( 1 + 1130 T^{4} - 10840461 T^{8} + 1130 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 - 7370 T^{4} + p^{4} T^{8} )^{2}( 1 + 7370 T^{4} + 40471059 T^{8} + 7370 p^{4} T^{12} + p^{8} T^{16} ) \) |
| 67 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{8} \) |
| 71 | \( ( 1 + 110 T^{2} + 7059 T^{4} + 110 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 - p T^{2} )^{16} \) |
| 79 | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{4}( 1 - 130 T^{2} + 10659 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 13130 T^{4} + 124938579 T^{8} + 13130 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 + p T^{2} )^{16} \) |
| 97 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.92569929434087272989408933253, −2.88926731272465208471697641122, −2.84982326597811260759921613293, −2.67781183079577348752917847655, −2.58483227389010513162667711710, −2.57135031665839240955436088606, −2.54189559958770775979578142921, −2.53638610167321929129841725018, −2.23302708061941482358736413481, −2.20855861217678739582443058093, −2.12841073081864353413478239169, −1.99483673214207478146575715363, −1.95991274780550477659788846774, −1.89549990871750761991194188534, −1.55279679069427808171358475570, −1.48016100437609017199346086120, −1.46509803086782960666644118065, −1.34099273667974788335563276012, −1.19282236489269154997024931278, −1.11611866935123965278639067781, −0.990806741210840406075961065132, −0.943821014965647007423445484886, −0.876640931442413628308369154133, −0.67940631140341181448159194360, −0.18260844250930219157382698140,
0.18260844250930219157382698140, 0.67940631140341181448159194360, 0.876640931442413628308369154133, 0.943821014965647007423445484886, 0.990806741210840406075961065132, 1.11611866935123965278639067781, 1.19282236489269154997024931278, 1.34099273667974788335563276012, 1.46509803086782960666644118065, 1.48016100437609017199346086120, 1.55279679069427808171358475570, 1.89549990871750761991194188534, 1.95991274780550477659788846774, 1.99483673214207478146575715363, 2.12841073081864353413478239169, 2.20855861217678739582443058093, 2.23302708061941482358736413481, 2.53638610167321929129841725018, 2.54189559958770775979578142921, 2.57135031665839240955436088606, 2.58483227389010513162667711710, 2.67781183079577348752917847655, 2.84982326597811260759921613293, 2.88926731272465208471697641122, 2.92569929434087272989408933253
Plot not available for L-functions of degree greater than 10.
