Among degree 4 primitive L-functions of analytic rank 2, this has the smallest conductor
| L(s) = 1 | − 3·2-s − 4·3-s + 4·4-s − 5·5-s + 12·6-s − 3·7-s − 3·8-s + 8·9-s + 15·10-s − 2·11-s − 16·12-s − 4·13-s + 9·14-s + 20·15-s + 3·16-s − 2·17-s − 24·18-s − 20·20-s + 12·21-s + 6·22-s − 23-s + 12·24-s + 12·25-s + 12·26-s − 12·27-s − 12·28-s − 8·29-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 2.30·3-s + 2·4-s − 2.23·5-s + 4.89·6-s − 1.13·7-s − 1.06·8-s + 8/3·9-s + 4.74·10-s − 0.603·11-s − 4.61·12-s − 1.10·13-s + 2.40·14-s + 5.16·15-s + 3/4·16-s − 0.485·17-s − 5.65·18-s − 4.47·20-s + 2.61·21-s + 1.27·22-s − 0.208·23-s + 2.44·24-s + 12/5·25-s + 2.35·26-s − 2.30·27-s − 2.26·28-s − 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3319 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3319 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3319 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 74 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + p T + 13 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 44 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 34 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T - p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 11 T + 65 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 103 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + T + 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 154 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 117 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 5 T + 26 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 32 T + 447 T^{2} + 32 p T^{3} + p^{2} T^{4} \) |
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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
−18.3938479658, −18.2580273292, −17.4645704872, −16.9879156301, −16.7814047905, −16.4058573769, −15.7955123936, −15.3310198499, −15.1226319001, −13.6247606441, −12.6576755659, −12.2114837333, −12.0193009042, −11.2005691286, −10.9981825876, −10.3483485727, −9.76512858602, −9.18509401409, −8.24396548524, −7.71994464966, −7.16377749696, −6.56064873395, −5.57423781475, −4.71287415515, −3.56807591826, 0, 0,
3.56807591826, 4.71287415515, 5.57423781475, 6.56064873395, 7.16377749696, 7.71994464966, 8.24396548524, 9.18509401409, 9.76512858602, 10.3483485727, 10.9981825876, 11.2005691286, 12.0193009042, 12.2114837333, 12.6576755659, 13.6247606441, 15.1226319001, 15.3310198499, 15.7955123936, 16.4058573769, 16.7814047905, 16.9879156301, 17.4645704872, 18.2580273292, 18.3938479658
