| L(s) = 1 | − 6·2-s − 2·3-s + 13·4-s + 10·5-s + 12·6-s + 12·8-s − 33·9-s − 60·10-s + 28·11-s − 26·12-s − 36·13-s − 20·15-s − 147·16-s − 76·17-s + 198·18-s + 160·19-s + 130·20-s − 168·22-s − 22·23-s − 24·24-s + 75·25-s + 216·26-s + 86·27-s − 250·29-s + 120·30-s − 132·31-s + 366·32-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 0.384·3-s + 13/8·4-s + 0.894·5-s + 0.816·6-s + 0.530·8-s − 1.22·9-s − 1.89·10-s + 0.767·11-s − 0.625·12-s − 0.768·13-s − 0.344·15-s − 2.29·16-s − 1.08·17-s + 2.59·18-s + 1.93·19-s + 1.45·20-s − 1.62·22-s − 0.199·23-s − 0.204·24-s + 3/5·25-s + 1.62·26-s + 0.612·27-s − 1.60·29-s + 0.730·30-s − 0.764·31-s + 2.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + 3 p T + 23 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 37 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 28 T + 2658 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 36 T + 4518 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 76 T + 5438 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 160 T + 18766 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 22 T - 2923 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 250 T + 57203 T^{2} + 250 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 132 T + 43938 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 416 T + 107578 T^{2} + 416 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 106 T + 138851 T^{2} + 106 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 666 T + 269853 T^{2} + 666 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 196 T + 209058 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 952 T + 484002 T^{2} + 952 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 840 T + 445646 T^{2} - 840 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 98 T - 193437 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1286 T + 1005453 T^{2} + 1286 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1064 T + 753846 T^{2} - 1064 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 172 T + 757582 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1240 T + 1278886 T^{2} + 1240 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1906 T + 2051733 T^{2} + 1906 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 650 T + 1305611 T^{2} - 650 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 628 T + 1423942 T^{2} + 628 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.16853512430119623386639145760, −10.88881867281334904105982150611, −10.01053091009945187870264282661, −9.932636763200759289914375614259, −9.473346156676402182437730040502, −9.028631295241562241997607439683, −8.614171235616486634737777594853, −8.328648720017293446535340096269, −7.45023945101520485217654847881, −7.18568175165495153682228229633, −6.57045099247808078447297762685, −5.86272575839997017268142487802, −5.15591631392687712811925272718, −4.95221504314018513809090339928, −3.72994095618257916756116375529, −2.94037788533811466017747117781, −1.79914287331358546503447521176, −1.45103348347805779905757510685, 0, 0,
1.45103348347805779905757510685, 1.79914287331358546503447521176, 2.94037788533811466017747117781, 3.72994095618257916756116375529, 4.95221504314018513809090339928, 5.15591631392687712811925272718, 5.86272575839997017268142487802, 6.57045099247808078447297762685, 7.18568175165495153682228229633, 7.45023945101520485217654847881, 8.328648720017293446535340096269, 8.614171235616486634737777594853, 9.028631295241562241997607439683, 9.473346156676402182437730040502, 9.932636763200759289914375614259, 10.01053091009945187870264282661, 10.88881867281334904105982150611, 11.16853512430119623386639145760
