| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s − 3·7-s + 8-s − 3·9-s + 2·10-s − 11-s + 12-s + 2·13-s + 3·14-s − 2·15-s − 3·16-s − 17-s + 3·18-s + 10·19-s − 2·20-s − 3·21-s + 22-s − 2·23-s + 24-s + 2·25-s − 2·26-s − 4·27-s − 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s − 9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s + 0.554·13-s + 0.801·14-s − 0.516·15-s − 3/4·16-s − 0.242·17-s + 0.707·18-s + 2.29·19-s − 0.447·20-s − 0.654·21-s + 0.213·22-s − 0.417·23-s + 0.204·24-s + 2/5·25-s − 0.392·26-s − 0.769·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1258 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1258 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4186305463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4186305463\) |
| \(L(\frac{3}{2})\) |
|
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_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 8 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + T - 24 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + T - 44 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 14 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 17 T + 158 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−19.6723362176, −19.2501534823, −18.4167238372, −18.2792818445, −17.4235402406, −16.6377673065, −16.1782316063, −15.9706871639, −15.2930038215, −14.5703144549, −13.8341811917, −13.4425284516, −12.6726718748, −11.7990588585, −11.3586905114, −10.7650901475, −9.76800435084, −9.34298692356, −8.57749615120, −7.81637322178, −7.37526280770, −6.37179526416, −5.33667934964, −3.77415576867, −2.90199334844,
2.90199334844, 3.77415576867, 5.33667934964, 6.37179526416, 7.37526280770, 7.81637322178, 8.57749615120, 9.34298692356, 9.76800435084, 10.7650901475, 11.3586905114, 11.7990588585, 12.6726718748, 13.4425284516, 13.8341811917, 14.5703144549, 15.2930038215, 15.9706871639, 16.1782316063, 16.6377673065, 17.4235402406, 18.2792818445, 18.4167238372, 19.2501534823, 19.6723362176
