| L(s) = 1 | + 2-s + 2·3-s − 2·5-s + 2·6-s + 7-s − 8-s + 3·9-s − 2·10-s − 3·11-s + 5·13-s + 14-s − 4·15-s − 16-s + 6·17-s + 3·18-s − 5·19-s + 2·21-s − 3·22-s − 2·24-s + 3·25-s + 5·26-s + 10·27-s − 4·30-s − 8·31-s − 6·33-s + 6·34-s − 2·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 0.894·5-s + 0.816·6-s + 0.377·7-s − 0.353·8-s + 9-s − 0.632·10-s − 0.904·11-s + 1.38·13-s + 0.267·14-s − 1.03·15-s − 1/4·16-s + 1.45·17-s + 0.707·18-s − 1.14·19-s + 0.436·21-s − 0.639·22-s − 0.408·24-s + 3/5·25-s + 0.980·26-s + 1.92·27-s − 0.730·30-s − 1.43·31-s − 1.04·33-s + 1.02·34-s − 0.338·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.913877912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.913877912\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−14.12784457836087530816329026197, −12.85814860192904584263256400795, −12.71965370516653741025565381520, −12.43652159705278810541361715704, −11.47308005407024246172270969987, −11.14897435120650597554620038914, −10.39801871047967570866772633202, −10.14577725609900815994866656798, −9.179995336337289743011884801363, −8.585720756094299345352879469530, −8.326706737784315408192174693614, −7.85652067569092072475132031267, −7.15664739908629145125579030445, −6.52583122783701339232766716257, −5.60566696672177544042946923934, −4.95083955912040390605254314489, −4.25266517359728745125292576278, −3.35015108045084868786478926176, −3.27272476255967007787113062498, −1.78280070726498445058305761751,
1.78280070726498445058305761751, 3.27272476255967007787113062498, 3.35015108045084868786478926176, 4.25266517359728745125292576278, 4.95083955912040390605254314489, 5.60566696672177544042946923934, 6.52583122783701339232766716257, 7.15664739908629145125579030445, 7.85652067569092072475132031267, 8.326706737784315408192174693614, 8.585720756094299345352879469530, 9.179995336337289743011884801363, 10.14577725609900815994866656798, 10.39801871047967570866772633202, 11.14897435120650597554620038914, 11.47308005407024246172270969987, 12.43652159705278810541361715704, 12.71965370516653741025565381520, 12.85814860192904584263256400795, 14.12784457836087530816329026197
