L(s) = 1 | − 2·2-s − 4·3-s + 4-s − 5·5-s + 8·6-s − 4·7-s + 2·8-s + 7·9-s + 10·10-s − 8·11-s − 4·12-s − 2·13-s + 8·14-s + 20·15-s − 3·16-s − 5·17-s − 14·18-s − 4·19-s − 5·20-s + 16·21-s + 16·22-s − 4·23-s − 8·24-s + 11·25-s + 4·26-s − 4·27-s − 4·28-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 2.30·3-s + 1/2·4-s − 2.23·5-s + 3.26·6-s − 1.51·7-s + 0.707·8-s + 7/3·9-s + 3.16·10-s − 2.41·11-s − 1.15·12-s − 0.554·13-s + 2.13·14-s + 5.16·15-s − 3/4·16-s − 1.21·17-s − 3.29·18-s − 0.917·19-s − 1.11·20-s + 3.49·21-s + 3.41·22-s − 0.834·23-s − 1.63·24-s + 11/5·25-s + 0.784·26-s − 0.769·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63506 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63506 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 113 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 15 T + p T^{2} ) \) |
| 281 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 18 T + p T^{2} ) \) |
good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 2 T - 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - T + 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 13 T + 82 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 86 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 138 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 148 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 80 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 118 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 15 T + 192 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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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
−15.5384483561, −15.1359090113, −14.5703050606, −13.3601872978, −13.1264258639, −12.6078932363, −12.5476794790, −11.6757964295, −11.4895314163, −10.9929212573, −10.7233242076, −10.2667182337, −9.87668164968, −9.24247470048, −8.47412634214, −8.16262648931, −7.59646445539, −7.13963509136, −6.80745878067, −5.90752845353, −5.48790627384, −4.90771238399, −4.17616136989, −3.63733356935, −2.44789168890, 0, 0, 0,
2.44789168890, 3.63733356935, 4.17616136989, 4.90771238399, 5.48790627384, 5.90752845353, 6.80745878067, 7.13963509136, 7.59646445539, 8.16262648931, 8.47412634214, 9.24247470048, 9.87668164968, 10.2667182337, 10.7233242076, 10.9929212573, 11.4895314163, 11.6757964295, 12.5476794790, 12.6078932363, 13.1264258639, 13.3601872978, 14.5703050606, 15.1359090113, 15.5384483561