| L(s) = 1 | − 2-s − 4-s + 4·7-s + 3·8-s − 2·9-s − 4·14-s − 16-s − 8·17-s + 2·18-s − 4·23-s − 6·25-s − 4·28-s + 7·31-s − 5·32-s + 8·34-s + 2·36-s + 4·41-s + 4·46-s + 4·47-s + 2·49-s + 6·50-s + 12·56-s − 7·62-s − 8·63-s + 7·64-s + 8·68-s + 8·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.51·7-s + 1.06·8-s − 2/3·9-s − 1.06·14-s − 1/4·16-s − 1.94·17-s + 0.471·18-s − 0.834·23-s − 6/5·25-s − 0.755·28-s + 1.25·31-s − 0.883·32-s + 1.37·34-s + 1/3·36-s + 0.624·41-s + 0.589·46-s + 0.583·47-s + 2/7·49-s + 0.848·50-s + 1.60·56-s − 0.889·62-s − 1.00·63-s + 7/8·64-s + 0.970·68-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4498002513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4498002513\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 8 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−13.58893806476612577003264141588, −12.74789436068042539706118437193, −11.90802956089560952645366515217, −11.24539587671841291284874296785, −11.00091428097796029987977540082, −10.14187008503575728910685183339, −9.435769140034684929745197900374, −8.736750895385704877224977334670, −8.184639727749500888957676415887, −7.82628569675387703973576419028, −6.74709665209108865989659837780, −5.70854502041029618274796868720, −4.74639531237434194774596723106, −4.14592737666349166326802179800, −2.11296470572071273256754525012,
2.11296470572071273256754525012, 4.14592737666349166326802179800, 4.74639531237434194774596723106, 5.70854502041029618274796868720, 6.74709665209108865989659837780, 7.82628569675387703973576419028, 8.184639727749500888957676415887, 8.736750895385704877224977334670, 9.435769140034684929745197900374, 10.14187008503575728910685183339, 11.00091428097796029987977540082, 11.24539587671841291284874296785, 11.90802956089560952645366515217, 12.74789436068042539706118437193, 13.58893806476612577003264141588
