| L(s) = 1 | − 2-s + 4-s − 2·5-s + 7-s − 8-s − 2·9-s + 2·10-s + 2·13-s − 14-s + 16-s − 6·17-s + 2·18-s − 2·20-s − 8·23-s + 2·25-s − 2·26-s + 28-s − 4·29-s + 4·31-s − 32-s + 6·34-s − 2·35-s − 2·36-s + 4·37-s + 2·40-s + 2·41-s − 16·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.632·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.471·18-s − 0.447·20-s − 1.66·23-s + 2/5·25-s − 0.392·26-s + 0.188·28-s − 0.742·29-s + 0.718·31-s − 0.176·32-s + 1.02·34-s − 0.338·35-s − 1/3·36-s + 0.657·37-s + 0.316·40-s + 0.312·41-s − 2.43·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T - 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 162 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 2 T - 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−15.2044887106, −14.7986331807, −14.2654895574, −13.7901197244, −13.2108905524, −12.9095825673, −11.9955195911, −11.8060432483, −11.3859674867, −11.0619071903, −10.4231508360, −10.0151603446, −9.30317318374, −8.83273717856, −8.27406452484, −8.08512942671, −7.50843087012, −6.74347751107, −6.34129777909, −5.70964625786, −4.86898551946, −4.19718886633, −3.57300806656, −2.65335980391, −1.72691978313, 0,
1.72691978313, 2.65335980391, 3.57300806656, 4.19718886633, 4.86898551946, 5.70964625786, 6.34129777909, 6.74347751107, 7.50843087012, 8.08512942671, 8.27406452484, 8.83273717856, 9.30317318374, 10.0151603446, 10.4231508360, 11.0619071903, 11.3859674867, 11.8060432483, 11.9955195911, 12.9095825673, 13.2108905524, 13.7901197244, 14.2654895574, 14.7986331807, 15.2044887106
