| L(s) = 1 | − 2-s − 2·4-s + 4·5-s − 2·7-s + 3·8-s − 4·10-s + 8·13-s + 2·14-s + 16-s + 6·19-s − 8·20-s − 8·23-s + 7·25-s − 8·26-s + 4·28-s + 6·29-s − 2·32-s − 8·35-s − 2·37-s − 6·38-s + 12·40-s − 12·41-s + 4·43-s + 8·46-s + 6·47-s + 3·49-s − 7·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 4-s + 1.78·5-s − 0.755·7-s + 1.06·8-s − 1.26·10-s + 2.21·13-s + 0.534·14-s + 1/4·16-s + 1.37·19-s − 1.78·20-s − 1.66·23-s + 7/5·25-s − 1.56·26-s + 0.755·28-s + 1.11·29-s − 0.353·32-s − 1.35·35-s − 0.328·37-s − 0.973·38-s + 1.89·40-s − 1.87·41-s + 0.609·43-s + 1.17·46-s + 0.875·47-s + 3/7·49-s − 0.989·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.381341024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.381341024\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 4 T + 9 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 107 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 - 16 T + 166 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 93 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 20 T + 241 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.054889660924422808795874581962, −8.048021290342914112249045118184, −7.33614294437518082727250367668, −6.98878561800577417141219446024, −6.49653195182888737768956535391, −6.31375161914847456974448985304, −6.01770258814134310729353519639, −5.60239389779301901110075338568, −5.37132773687808546506323146494, −5.10062856065806248588833125943, −4.32029451758007965253316838943, −4.24230112488293483571467343955, −3.65486463020107708558427474557, −3.34848217830436975198088244269, −2.95388986901501374464163272196, −2.34574996071983003778520550714, −1.81240211855875042326547033886, −1.49227785884528830042038683493, −0.921392421691580142330643156260, −0.52555158065071403280017369690,
0.52555158065071403280017369690, 0.921392421691580142330643156260, 1.49227785884528830042038683493, 1.81240211855875042326547033886, 2.34574996071983003778520550714, 2.95388986901501374464163272196, 3.34848217830436975198088244269, 3.65486463020107708558427474557, 4.24230112488293483571467343955, 4.32029451758007965253316838943, 5.10062856065806248588833125943, 5.37132773687808546506323146494, 5.60239389779301901110075338568, 6.01770258814134310729353519639, 6.31375161914847456974448985304, 6.49653195182888737768956535391, 6.98878561800577417141219446024, 7.33614294437518082727250367668, 8.048021290342914112249045118184, 8.054889660924422808795874581962
