| L(s) = 1 | − 2·3-s + 4·7-s + 3·9-s + 2·11-s + 2·13-s − 4·17-s + 4·19-s − 8·21-s − 4·23-s − 8·25-s − 4·27-s − 4·29-s − 4·31-s − 4·33-s − 12·37-s − 4·39-s − 8·41-s − 8·43-s + 8·47-s − 2·49-s + 8·51-s + 4·53-s − 8·57-s + 8·59-s − 20·61-s + 12·63-s − 4·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.51·7-s + 9-s + 0.603·11-s + 0.554·13-s − 0.970·17-s + 0.917·19-s − 1.74·21-s − 0.834·23-s − 8/5·25-s − 0.769·27-s − 0.742·29-s − 0.718·31-s − 0.696·33-s − 1.97·37-s − 0.640·39-s − 1.24·41-s − 1.21·43-s + 1.16·47-s − 2/7·49-s + 1.12·51-s + 0.549·53-s − 1.05·57-s + 1.04·59-s − 2.56·61-s + 1.51·63-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47114496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47114496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 36 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 84 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 156 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 176 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.56955193019593057977621598347, −7.48893076607711952478434636659, −6.96221359835710091395249928973, −6.93642667619504502594997579755, −6.16220906336544668737446077761, −6.09498411682453723189603741944, −5.53971399831337730228430306530, −5.46004483086053394164649244294, −4.97578480432390573630353237433, −4.61475860525058631689538374708, −4.18737295292234124095553376140, −4.03950769278670799724098243709, −3.31295414611217231645596741628, −3.25630609288498386305117751841, −2.12900716040407535598434995791, −1.91075479554379781913261339442, −1.51849688131211854983026299909, −1.22343805985830353880684867366, 0, 0,
1.22343805985830353880684867366, 1.51849688131211854983026299909, 1.91075479554379781913261339442, 2.12900716040407535598434995791, 3.25630609288498386305117751841, 3.31295414611217231645596741628, 4.03950769278670799724098243709, 4.18737295292234124095553376140, 4.61475860525058631689538374708, 4.97578480432390573630353237433, 5.46004483086053394164649244294, 5.53971399831337730228430306530, 6.09498411682453723189603741944, 6.16220906336544668737446077761, 6.93642667619504502594997579755, 6.96221359835710091395249928973, 7.48893076607711952478434636659, 7.56955193019593057977621598347
