| L(s) = 1 | − 32·25-s + 32·37-s − 32·43-s − 32·67-s − 32·109-s + 72·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 64·169-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 | − 6.39·25-s + 5.26·37-s − 4.87·43-s − 3.90·67-s − 3.06·109-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 + 4.92·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.861398901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.861398901\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 + 16 T^{2} + 112 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 32 T^{2} + 496 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 32 T^{2} + 672 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 60 T^{2} + 1590 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 4 T^{2} - 90 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 68 T^{2} + 2326 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 44 T^{2} + 2374 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 16 T^{2} - 992 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 20 T^{2} + 2950 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 48 T^{2} + 1586 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 36 T^{2} + 5718 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 80 T^{2} + 4240 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 20 T^{2} + 8134 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 144 T^{2} + 11424 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 204 T^{2} + 22134 T^{4} + 204 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 144 T^{2} + 10368 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 240 T^{2} + 28800 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.18732767596628213091442436860, −3.18459500820480824902242495542, −3.05728375841168348549887866496, −2.94129752733538838930472193053, −2.73788075551882949494270885236, −2.69329154210325933276248202910, −2.54610498456801539280325423888, −2.45863979621409175291418048638, −2.29093870557043999334601351219, −2.14238445368566560764973976741, −2.02652636946642069654432699520, −1.95530772859399480074207728494, −1.91672715412326716306449515307, −1.86252699715047978992491251418, −1.73183435099995322783380350634, −1.40522048701305635593666524664, −1.39160303264584638435960655544, −1.22338871185203330208887761231, −1.19650292133054703880367955097, −1.11048373262862587614563108690, −0.68721712199752029433449252416, −0.48726836870764441151489964093, −0.34085115419983131679941007933, −0.27825117737159696435732776271, −0.26381859171003154244324912129,
0.26381859171003154244324912129, 0.27825117737159696435732776271, 0.34085115419983131679941007933, 0.48726836870764441151489964093, 0.68721712199752029433449252416, 1.11048373262862587614563108690, 1.19650292133054703880367955097, 1.22338871185203330208887761231, 1.39160303264584638435960655544, 1.40522048701305635593666524664, 1.73183435099995322783380350634, 1.86252699715047978992491251418, 1.91672715412326716306449515307, 1.95530772859399480074207728494, 2.02652636946642069654432699520, 2.14238445368566560764973976741, 2.29093870557043999334601351219, 2.45863979621409175291418048638, 2.54610498456801539280325423888, 2.69329154210325933276248202910, 2.73788075551882949494270885236, 2.94129752733538838930472193053, 3.05728375841168348549887866496, 3.18459500820480824902242495542, 3.18732767596628213091442436860
Plot not available for L-functions of degree greater than 10.
