| L(s) = 1 | − 2-s + 3-s − 4-s − 2·5-s − 6-s + 3·7-s + 8-s + 2·10-s − 2·11-s − 12-s − 3·14-s − 2·15-s + 3·16-s + 6·17-s − 9·19-s + 2·20-s + 3·21-s + 2·22-s − 23-s + 24-s − 2·25-s + 2·27-s − 3·28-s − 4·29-s + 2·30-s − 6·31-s − 3·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s + 0.632·10-s − 0.603·11-s − 0.288·12-s − 0.801·14-s − 0.516·15-s + 3/4·16-s + 1.45·17-s − 2.06·19-s + 0.447·20-s + 0.654·21-s + 0.426·22-s − 0.208·23-s + 0.204·24-s − 2/5·25-s + 0.384·27-s − 0.566·28-s − 0.742·29-s + 0.365·30-s − 1.07·31-s − 0.530·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81962 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81962 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 107 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 8 T + p T^{2} ) \) |
| 383 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 21 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 9 T + 3 p T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 31 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 7 T + 72 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 9 T + 85 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7 T + 82 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T + 106 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 154 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 46 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 7 T + 144 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
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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
−14.6100554003, −14.2679979525, −13.5509766117, −13.1442787682, −12.7344998106, −12.1807103213, −11.8455554273, −11.2283526253, −10.8677538366, −10.3223844888, −9.88868132110, −9.48254002785, −8.64102246123, −8.38475027877, −8.22643932041, −7.77888054057, −7.20395692219, −6.58297195300, −5.56902551173, −5.30766291397, −4.56548325714, −3.82519899404, −3.53198193517, −2.40660777174, −1.53576437070, 0,
1.53576437070, 2.40660777174, 3.53198193517, 3.82519899404, 4.56548325714, 5.30766291397, 5.56902551173, 6.58297195300, 7.20395692219, 7.77888054057, 8.22643932041, 8.38475027877, 8.64102246123, 9.48254002785, 9.88868132110, 10.3223844888, 10.8677538366, 11.2283526253, 11.8455554273, 12.1807103213, 12.7344998106, 13.1442787682, 13.5509766117, 14.2679979525, 14.6100554003
