| L(s) = 1 | − 4-s − 2·5-s + 3·7-s − 2·8-s − 2·9-s − 3·11-s + 16-s + 17-s + 5·19-s + 2·20-s − 5·23-s + 2·25-s − 3·28-s + 4·29-s − 2·31-s + 4·32-s − 6·35-s + 2·36-s + 2·37-s + 4·40-s + 3·41-s + 3·44-s + 4·45-s + 2·47-s − 49-s + 9·53-s + 6·55-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.894·5-s + 1.13·7-s − 0.707·8-s − 2/3·9-s − 0.904·11-s + 1/4·16-s + 0.242·17-s + 1.14·19-s + 0.447·20-s − 1.04·23-s + 2/5·25-s − 0.566·28-s + 0.742·29-s − 0.359·31-s + 0.707·32-s − 1.01·35-s + 1/3·36-s + 0.328·37-s + 0.632·40-s + 0.468·41-s + 0.452·44-s + 0.596·45-s + 0.291·47-s − 1/7·49-s + 1.23·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1270 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1270 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4736509421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4736509421\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| 127 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 20 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 26 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 38 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−19.6568311241, −18.9503890316, −18.3576321607, −17.9939273287, −17.6312819661, −16.8834354892, −16.0779288654, −15.7672744879, −14.9597293671, −14.6061765040, −13.9435487587, −13.4462724835, −12.4378997441, −11.9874616952, −11.4861866002, −10.8390478871, −10.0245475085, −9.20293686385, −8.32131772067, −8.04671447301, −7.28147039781, −5.89944013519, −5.20556006040, −4.23659828419, −2.97079164477,
2.97079164477, 4.23659828419, 5.20556006040, 5.89944013519, 7.28147039781, 8.04671447301, 8.32131772067, 9.20293686385, 10.0245475085, 10.8390478871, 11.4861866002, 11.9874616952, 12.4378997441, 13.4462724835, 13.9435487587, 14.6061765040, 14.9597293671, 15.7672744879, 16.0779288654, 16.8834354892, 17.6312819661, 17.9939273287, 18.3576321607, 18.9503890316, 19.6568311241
