| L(s) = 1 | − 2·3-s + 4-s + 9-s − 2·12-s + 5·13-s + 16-s − 3·17-s − 6·23-s + 8·25-s + 4·27-s − 9·29-s + 36-s − 10·39-s − 11·43-s − 2·48-s + 49-s + 6·51-s + 5·52-s − 21·53-s − 2·61-s + 64-s − 3·68-s + 12·69-s − 16·75-s − 11·79-s − 11·81-s + 18·87-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 1/3·9-s − 0.577·12-s + 1.38·13-s + 1/4·16-s − 0.727·17-s − 1.25·23-s + 8/5·25-s + 0.769·27-s − 1.67·29-s + 1/6·36-s − 1.60·39-s − 1.67·43-s − 0.288·48-s + 1/7·49-s + 0.840·51-s + 0.693·52-s − 2.88·53-s − 0.256·61-s + 1/8·64-s − 0.363·68-s + 1.44·69-s − 1.84·75-s − 1.23·79-s − 1.22·81-s + 1.92·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 149 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
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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
−8.673398679322317121078093038242, −8.163271224594245044011765020481, −7.57010044012870732497075612996, −7.08907078839956414344875232548, −6.48828034173872269887290650938, −6.20516520488743930407219198497, −5.94956153602807447614023709668, −5.19196318896285656930006115107, −4.85225003171786539733986973099, −4.16010528940181342441065299572, −3.50235509909040990057066926894, −2.97304437844994934858034201686, −1.95875131017253697759164380005, −1.31599634754427426074556227646, 0,
1.31599634754427426074556227646, 1.95875131017253697759164380005, 2.97304437844994934858034201686, 3.50235509909040990057066926894, 4.16010528940181342441065299572, 4.85225003171786539733986973099, 5.19196318896285656930006115107, 5.94956153602807447614023709668, 6.20516520488743930407219198497, 6.48828034173872269887290650938, 7.08907078839956414344875232548, 7.57010044012870732497075612996, 8.163271224594245044011765020481, 8.673398679322317121078093038242
