| L(s) = 1 | − 8·2-s + 36·4-s − 120·8-s − 8·11-s + 330·16-s + 64·22-s − 4·23-s + 16·25-s − 4·29-s − 792·32-s + 32·37-s + 28·43-s − 288·44-s + 32·46-s − 128·50-s − 20·53-s + 32·58-s + 1.71e3·64-s − 72·67-s − 48·71-s − 256·74-s + 16·79-s + 9·81-s − 224·86-s + 960·88-s − 144·92-s + 576·100-s + ⋯ |
| L(s) = 1 | − 5.65·2-s + 18·4-s − 42.4·8-s − 2.41·11-s + 82.5·16-s + 13.6·22-s − 0.834·23-s + 16/5·25-s − 0.742·29-s − 140.·32-s + 5.26·37-s + 4.26·43-s − 43.4·44-s + 4.71·46-s − 18.1·50-s − 2.74·53-s + 4.20·58-s + 214.5·64-s − 8.79·67-s − 5.69·71-s − 29.7·74-s + 1.80·79-s + 81-s − 24.1·86-s + 102.·88-s − 15.0·92-s + 57.5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06984646392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06984646392\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T )^{8} \) |
| 3 | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 16 T^{2} + 29 p T^{4} - 976 T^{6} + 5296 T^{8} - 976 p^{2} T^{10} + 29 p^{5} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 4 T + 2 T^{2} - 32 T^{3} - 101 T^{4} - 32 p T^{5} + 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 20 T^{2} + 231 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 16 T^{2} - 194 T^{4} + 2048 T^{6} + 44995 T^{8} + 2048 p^{2} T^{10} - 194 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 - 72 T^{2} + 3169 T^{4} - 93096 T^{6} + 2058480 T^{8} - 93096 p^{2} T^{10} + 3169 p^{4} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 + 2 T + 5 T^{2} - 94 T^{3} - 620 T^{4} - 94 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 2 T - 52 T^{2} - 4 T^{3} + 2179 T^{4} - 4 p T^{5} - 52 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 50 T^{2} + 819 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 14 T + 88 T^{2} - 308 T^{3} + 1387 T^{4} - 308 p T^{5} + 88 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 76 T^{2} + 4662 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 10 T - 28 T^{2} + 220 T^{3} + 8275 T^{4} + 220 p T^{5} - 28 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 8 T^{2} - 2430 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 216 T^{2} + 19079 T^{4} + 216 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 18 T + 188 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 12 T + 151 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 - 180 T^{2} + 14842 T^{4} - 1242000 T^{6} + 107225523 T^{8} - 1242000 p^{2} T^{10} + 14842 p^{4} T^{12} - 180 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 68 T^{2} - 2265 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 + 80 T^{2} - 9314 T^{4} - 10240 T^{6} + 133493155 T^{8} - 10240 p^{2} T^{10} - 9314 p^{4} T^{12} + 80 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 - 372 T^{2} + 85018 T^{4} - 12851856 T^{6} + 1457345619 T^{8} - 12851856 p^{2} T^{10} + 85018 p^{4} T^{12} - 372 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−4.51612564643574825928855110575, −4.30424711503623088028217870070, −4.09051029625363667770854670515, −3.80657966632598295180111133303, −3.77122631588711922113128629313, −3.51822023304080917422853131159, −3.14502808738321113603691671305, −3.09262777402254045510598126952, −2.97426478016247542695006481187, −2.84803711208535615590004090211, −2.82291491530787779788584744884, −2.72796142768427488678440137426, −2.60306195195237913732946795167, −2.56809068636433414786694054518, −2.25614119793965932513469746377, −2.01268948403321958875719025800, −1.77474860615799703809459296091, −1.74046208870896397572591369528, −1.48740179396304525610555277719, −1.35733940898393730287615998155, −1.12493508923873528824968270482, −0.912002545428551141285855067777, −0.69178018853338750252696877068, −0.45682979462238907134217828964, −0.16408342159455284089235721150,
0.16408342159455284089235721150, 0.45682979462238907134217828964, 0.69178018853338750252696877068, 0.912002545428551141285855067777, 1.12493508923873528824968270482, 1.35733940898393730287615998155, 1.48740179396304525610555277719, 1.74046208870896397572591369528, 1.77474860615799703809459296091, 2.01268948403321958875719025800, 2.25614119793965932513469746377, 2.56809068636433414786694054518, 2.60306195195237913732946795167, 2.72796142768427488678440137426, 2.82291491530787779788584744884, 2.84803711208535615590004090211, 2.97426478016247542695006481187, 3.09262777402254045510598126952, 3.14502808738321113603691671305, 3.51822023304080917422853131159, 3.77122631588711922113128629313, 3.80657966632598295180111133303, 4.09051029625363667770854670515, 4.30424711503623088028217870070, 4.51612564643574825928855110575
Plot not available for L-functions of degree greater than 10.
