| L(s) = 1 | + 2·3-s − 4-s − 2·9-s − 2·12-s − 2·13-s − 3·16-s − 2·17-s + 3·23-s + 6·25-s − 10·27-s + 8·29-s + 2·36-s − 4·39-s + 10·43-s − 6·48-s − 8·49-s − 4·51-s + 2·52-s − 8·53-s + 7·64-s + 2·68-s + 6·69-s + 12·75-s − 14·79-s − 5·81-s + 16·87-s − 3·92-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s − 2/3·9-s − 0.577·12-s − 0.554·13-s − 3/4·16-s − 0.485·17-s + 0.625·23-s + 6/5·25-s − 1.92·27-s + 1.48·29-s + 1/3·36-s − 0.640·39-s + 1.52·43-s − 0.866·48-s − 8/7·49-s − 0.560·51-s + 0.277·52-s − 1.09·53-s + 7/8·64-s + 0.242·68-s + 0.722·69-s + 1.38·75-s − 1.57·79-s − 5/9·81-s + 1.71·87-s − 0.312·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3887 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3887 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9009011983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9009011983\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 170 T^{2} + p^{2} T^{4} \) |
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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
−12.73533774163262736763912873714, −11.99194214963161824650437188889, −11.26475990806612486488139835344, −10.84758190365504273016954464430, −9.929652181218310602263282178320, −9.311341964711249074616543068169, −8.781296642082660196397410944930, −8.496631523612372874930816564504, −7.72629502163803240846402078344, −6.92983547406076020553289617152, −6.13771170368324671342892193474, −5.06169023374393873111537198259, −4.42013484455619515379930105876, −3.17090161637365476303900486433, −2.51569256430844859340267012894,
2.51569256430844859340267012894, 3.17090161637365476303900486433, 4.42013484455619515379930105876, 5.06169023374393873111537198259, 6.13771170368324671342892193474, 6.92983547406076020553289617152, 7.72629502163803240846402078344, 8.496631523612372874930816564504, 8.781296642082660196397410944930, 9.311341964711249074616543068169, 9.929652181218310602263282178320, 10.84758190365504273016954464430, 11.26475990806612486488139835344, 11.99194214963161824650437188889, 12.73533774163262736763912873714
