| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s − 2·11-s + 12-s − 2·13-s − 16-s − 3·17-s + 18-s − 7·19-s − 2·22-s − 3·23-s + 3·24-s − 5·25-s − 2·26-s − 27-s + 10·29-s − 3·31-s + 5·32-s + 2·33-s − 3·34-s − 36-s + 2·37-s − 7·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s − 0.554·13-s − 1/4·16-s − 0.727·17-s + 0.235·18-s − 1.60·19-s − 0.426·22-s − 0.625·23-s + 0.612·24-s − 25-s − 0.392·26-s − 0.192·27-s + 1.85·29-s − 0.538·31-s + 0.883·32-s + 0.348·33-s − 0.514·34-s − 1/6·36-s + 0.328·37-s − 1.13·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T - 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 76 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 30 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 68 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 28 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 120 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 48 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 142 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 66 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 160 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−15.3690900110, −14.9699931798, −14.4798303230, −13.9798893888, −13.5758196096, −12.9898737826, −12.8176508768, −12.0980642757, −12.0010256533, −11.2320204030, −10.7302814836, −10.2349186491, −9.77495574274, −9.18408920509, −8.57570071976, −8.09333518504, −7.53635369876, −6.53880073609, −6.34419170344, −5.69608978943, −4.90077466849, −4.56953755097, −3.97202210850, −2.99626304388, −2.08233427794, 0,
2.08233427794, 2.99626304388, 3.97202210850, 4.56953755097, 4.90077466849, 5.69608978943, 6.34419170344, 6.53880073609, 7.53635369876, 8.09333518504, 8.57570071976, 9.18408920509, 9.77495574274, 10.2349186491, 10.7302814836, 11.2320204030, 12.0010256533, 12.0980642757, 12.8176508768, 12.9898737826, 13.5758196096, 13.9798893888, 14.4798303230, 14.9699931798, 15.3690900110
