| L(s) = 1 | + 4·4-s + 6·16-s + 8·121-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 + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
| L(s) = 1 | + 4·4-s + 6·16-s + 8·121-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 + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.081631826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.081631826\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 11 | \( ( 1 - T^{2} + T^{4} )^{8} \) |
| 13 | \( ( 1 + T^{8} )^{4} \) |
| 17 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 19 | \( ( 1 - T^{2} + T^{4} )^{8} \) |
| 23 | \( ( 1 - T^{2} + T^{4} )^{8} \) |
| 29 | \( ( 1 - T )^{16}( 1 + T )^{16} \) |
| 31 | \( ( 1 - T^{2} + T^{4} )^{8} \) |
| 37 | \( ( 1 - T^{4} + T^{8} )^{4} \) |
| 41 | \( ( 1 + T^{8} )^{4} \) |
| 43 | \( ( 1 - T )^{16}( 1 + T )^{16} \) |
| 47 | \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \) |
| 53 | \( ( 1 - T^{4} + T^{8} )^{4} \) |
| 59 | \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \) |
| 61 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 67 | \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \) |
| 71 | \( ( 1 + T^{2} )^{16} \) |
| 73 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 79 | \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \) |
| 83 | \( ( 1 - T )^{16}( 1 + T )^{16} \) |
| 89 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 97 | \( ( 1 + T^{8} )^{4} \) |
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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
−2.53411694799619878126008621084, −2.49490116475963109977274232561, −2.47028648725346020480655871121, −2.36321983537022563119375797175, −2.36284917591724346278549527624, −2.32722492502241117114289205460, −2.14780905943798996472039927282, −2.10166551084085750235790744150, −2.09158838860886240809228948874, −1.88203065044602014363585955014, −1.81146344584377180028351987538, −1.77473474852790637017936914095, −1.76214967257629880486954558777, −1.62470278696969808446309036050, −1.55757806277339257792812508640, −1.54574740444573382898579389523, −1.54072696811007900883727946681, −1.39456597428294982672270065276, −1.34426446403553241771102251631, −0.986698763097903668707774752801, −0.930240844354372354396917701343, −0.877955672697376951520529859658, −0.857297185712590379902205919648, −0.67258309858569759672237979264, −0.44664890100557229653747202845,
0.44664890100557229653747202845, 0.67258309858569759672237979264, 0.857297185712590379902205919648, 0.877955672697376951520529859658, 0.930240844354372354396917701343, 0.986698763097903668707774752801, 1.34426446403553241771102251631, 1.39456597428294982672270065276, 1.54072696811007900883727946681, 1.54574740444573382898579389523, 1.55757806277339257792812508640, 1.62470278696969808446309036050, 1.76214967257629880486954558777, 1.77473474852790637017936914095, 1.81146344584377180028351987538, 1.88203065044602014363585955014, 2.09158838860886240809228948874, 2.10166551084085750235790744150, 2.14780905943798996472039927282, 2.32722492502241117114289205460, 2.36284917591724346278549527624, 2.36321983537022563119375797175, 2.47028648725346020480655871121, 2.49490116475963109977274232561, 2.53411694799619878126008621084
Plot not available for L-functions of degree greater than 10.
