| L(s) = 1 | + 2-s − 3·3-s − 2·4-s + 2·5-s − 3·6-s − 3·7-s − 3·8-s + 2·9-s + 2·10-s + 10·11-s + 6·12-s − 2·13-s − 3·14-s − 6·15-s + 16-s + 4·17-s + 2·18-s − 12·19-s − 4·20-s + 9·21-s + 10·22-s − 8·23-s + 9·24-s − 2·25-s − 2·26-s + 6·27-s + 6·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s − 4-s + 0.894·5-s − 1.22·6-s − 1.13·7-s − 1.06·8-s + 2/3·9-s + 0.632·10-s + 3.01·11-s + 1.73·12-s − 0.554·13-s − 0.801·14-s − 1.54·15-s + 1/4·16-s + 0.970·17-s + 0.471·18-s − 2.75·19-s − 0.894·20-s + 1.96·21-s + 2.13·22-s − 1.66·23-s + 1.83·24-s − 2/5·25-s − 0.392·26-s + 1.15·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64754209 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64754209 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.138110062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.138110062\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| 619 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 12 T + 69 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 73 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 81 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 14 T + 130 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 17 T + 155 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 11 T + 117 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 51 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 163 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 177 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 7 T + 179 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 19 T + 253 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.160808142820660269042076563662, −7.51765719916662807588965463678, −6.78539019149311761263797141003, −6.70378590707603140027602997971, −6.53047568122544348573481725286, −6.18954150855916691256307382640, −5.85944248714376720987578440052, −5.72446600462784541316446902832, −5.32172201460189161230890811826, −4.94258001466775102999026884635, −4.25175162268397730553425788011, −4.20241811247347596179618538660, −4.03306415257982623498696513385, −3.59427422183073332827817802558, −2.99403507139816978495714122321, −2.51826035598364231014167048931, −1.71092222731916602807218821990, −1.65837434190666035108952209561, −0.58497649156770714201763802710, −0.44544847473242977688641858282,
0.44544847473242977688641858282, 0.58497649156770714201763802710, 1.65837434190666035108952209561, 1.71092222731916602807218821990, 2.51826035598364231014167048931, 2.99403507139816978495714122321, 3.59427422183073332827817802558, 4.03306415257982623498696513385, 4.20241811247347596179618538660, 4.25175162268397730553425788011, 4.94258001466775102999026884635, 5.32172201460189161230890811826, 5.72446600462784541316446902832, 5.85944248714376720987578440052, 6.18954150855916691256307382640, 6.53047568122544348573481725286, 6.70378590707603140027602997971, 6.78539019149311761263797141003, 7.51765719916662807588965463678, 8.160808142820660269042076563662
