| L(s) = 1 | − 2-s − 4-s − 5-s + 3·8-s − 4·9-s + 10-s − 6·11-s − 16-s − 2·17-s + 4·18-s − 2·19-s + 20-s + 6·22-s + 6·23-s − 4·25-s − 6·29-s − 3·31-s − 5·32-s + 2·34-s + 4·36-s + 5·37-s + 2·38-s − 3·40-s − 3·41-s + 6·44-s + 4·45-s − 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s − 4/3·9-s + 0.316·10-s − 1.80·11-s − 1/4·16-s − 0.485·17-s + 0.942·18-s − 0.458·19-s + 0.223·20-s + 1.27·22-s + 1.25·23-s − 4/5·25-s − 1.11·29-s − 0.538·31-s − 0.883·32-s + 0.342·34-s + 2/3·36-s + 0.821·37-s + 0.324·38-s − 0.474·40-s − 0.468·41-s + 0.904·44-s + 0.596·45-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10040 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10040 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 251 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 26 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 24 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 40 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 110 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 132 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 110 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 160 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 11 T + 124 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−16.8687747396, −16.5447652818, −15.9336384998, −15.3031811085, −14.9950329496, −14.4328372796, −13.7164154962, −13.3646133883, −12.9227433333, −12.4250752079, −11.4912343006, −11.1001033091, −10.7749931823, −10.1222680444, −9.39581553983, −9.01348583108, −8.19468738670, −8.07573189962, −7.42908562490, −6.60650478528, −5.48316769767, −5.27800992351, −4.31947208015, −3.32351834016, −2.31959028391, 0,
2.31959028391, 3.32351834016, 4.31947208015, 5.27800992351, 5.48316769767, 6.60650478528, 7.42908562490, 8.07573189962, 8.19468738670, 9.01348583108, 9.39581553983, 10.1222680444, 10.7749931823, 11.1001033091, 11.4912343006, 12.4250752079, 12.9227433333, 13.3646133883, 13.7164154962, 14.4328372796, 14.9950329496, 15.3031811085, 15.9336384998, 16.5447652818, 16.8687747396
