| L(s) = 1 | − 2-s + 2·5-s + 7-s + 8-s − 2·10-s + 11-s + 6·13-s − 14-s − 16-s + 10·17-s − 14·19-s − 22-s + 4·23-s + 5·25-s − 6·26-s − 4·29-s + 6·31-s − 10·34-s + 2·35-s + 4·37-s + 14·38-s + 2·40-s + 3·41-s + 43-s − 4·46-s − 5·50-s − 24·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.894·5-s + 0.377·7-s + 0.353·8-s − 0.632·10-s + 0.301·11-s + 1.66·13-s − 0.267·14-s − 1/4·16-s + 2.42·17-s − 3.21·19-s − 0.213·22-s + 0.834·23-s + 25-s − 1.17·26-s − 0.742·29-s + 1.07·31-s − 1.71·34-s + 0.338·35-s + 0.657·37-s + 2.27·38-s + 0.316·40-s + 0.468·41-s + 0.152·43-s − 0.589·46-s − 0.707·50-s − 3.29·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.517818456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.517818456\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 16 T + 173 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−11.21382476913412426416821379411, −11.01664605863384974468985906339, −10.66320997310013936672319637743, −10.28490712116099622030886667793, −9.649564808002223258409360373451, −9.361017597321515232368963967865, −8.786933477220466146940181912790, −8.478132158562124749100867162210, −7.82975644487966316227544025438, −7.79682463233054084338892125783, −6.64664905897367195457926409161, −6.22409520970669069439154576757, −6.19279108627659264670137224198, −5.25799444453769246158374559520, −4.79080207239947102306857522697, −4.01793267869358109412039514742, −3.47606948117361845475037224360, −2.58001998311368792871577853275, −1.67614299239604350579256902458, −1.07566152884837080457345135632,
1.07566152884837080457345135632, 1.67614299239604350579256902458, 2.58001998311368792871577853275, 3.47606948117361845475037224360, 4.01793267869358109412039514742, 4.79080207239947102306857522697, 5.25799444453769246158374559520, 6.19279108627659264670137224198, 6.22409520970669069439154576757, 6.64664905897367195457926409161, 7.79682463233054084338892125783, 7.82975644487966316227544025438, 8.478132158562124749100867162210, 8.786933477220466146940181912790, 9.361017597321515232368963967865, 9.649564808002223258409360373451, 10.28490712116099622030886667793, 10.66320997310013936672319637743, 11.01664605863384974468985906339, 11.21382476913412426416821379411
