| L(s) = 1 | − 3-s − 4-s − 5·5-s − 2·8-s − 9-s − 4·11-s + 12-s − 5·13-s + 5·15-s + 16-s − 17-s + 5·20-s − 7·23-s + 2·24-s + 10·25-s − 6·29-s + 3·31-s + 4·32-s + 4·33-s + 36-s + 2·37-s + 5·39-s + 10·40-s − 3·41-s + 43-s + 4·44-s + 5·45-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s − 2.23·5-s − 0.707·8-s − 1/3·9-s − 1.20·11-s + 0.288·12-s − 1.38·13-s + 1.29·15-s + 1/4·16-s − 0.242·17-s + 1.11·20-s − 1.45·23-s + 0.408·24-s + 2·25-s − 1.11·29-s + 0.538·31-s + 0.707·32-s + 0.696·33-s + 1/6·36-s + 0.328·37-s + 0.800·39-s + 1.58·40-s − 0.468·41-s + 0.152·43-s + 0.603·44-s + 0.745·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68158 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68158 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 7 T + p T^{2} ) \) |
| 643 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 36 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_4$ | \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 45 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3 T + 43 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 72 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T - 36 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 51 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 89 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 90 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 108 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T + 44 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 3 T + 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - T + 73 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−15.0518415423, −14.6377819289, −14.0626308825, −13.3980231782, −13.0148997674, −12.3863133120, −12.1790639980, −11.6598981713, −11.5099801843, −11.0290257390, −10.3000401222, −9.90568734237, −9.44989737301, −8.56628282054, −8.32502641061, −7.75704593837, −7.54599461266, −6.91719089117, −6.10255770260, −5.53840530742, −4.95866871271, −4.39541407745, −3.84292562234, −3.17999097637, −2.37238899921, 0, 0,
2.37238899921, 3.17999097637, 3.84292562234, 4.39541407745, 4.95866871271, 5.53840530742, 6.10255770260, 6.91719089117, 7.54599461266, 7.75704593837, 8.32502641061, 8.56628282054, 9.44989737301, 9.90568734237, 10.3000401222, 11.0290257390, 11.5099801843, 11.6598981713, 12.1790639980, 12.3863133120, 13.0148997674, 13.3980231782, 14.0626308825, 14.6377819289, 15.0518415423
