| L(s) = 1 | + 4·2-s − 52·4-s − 352·8-s + 568·11-s + 1.61e3·16-s + 2.27e3·22-s − 2.99e3·23-s − 634·25-s + 8.73e3·29-s + 1.96e4·32-s − 2.52e4·37-s − 2.71e3·43-s − 2.95e4·44-s − 1.19e4·46-s − 2.53e3·50-s − 2.83e4·53-s + 3.49e4·58-s − 2.31e4·64-s − 7.28e3·67-s − 7.12e4·71-s − 1.01e5·74-s − 1.09e5·79-s − 1.08e4·86-s − 1.99e5·88-s + 1.55e5·92-s + 3.29e4·100-s − 1.13e5·106-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.62·4-s − 1.94·8-s + 1.41·11-s + 1.57·16-s + 1.00·22-s − 1.17·23-s − 0.202·25-s + 1.92·29-s + 3.39·32-s − 3.03·37-s − 0.223·43-s − 2.29·44-s − 0.833·46-s − 0.143·50-s − 1.38·53-s + 1.36·58-s − 0.705·64-s − 0.198·67-s − 1.67·71-s − 2.14·74-s − 1.96·79-s − 0.158·86-s − 2.75·88-s + 1.91·92-s + 0.329·100-s − 0.978·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p^{5} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 634 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 284 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 35954 p T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2817250 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 229142 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 1496 T + p^{5} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4366 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 15722366 T^{2} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 12630 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 142552786 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 1356 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 357849118 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 14150 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 31458982 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 422082602 T^{2} + p^{10} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 3644 T + p^{5} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 35632 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2484166610 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 54616 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 7877806102 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 10752624754 T^{2} + p^{10} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 16535359486 T^{2} + p^{10} T^{4} \) |
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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
−9.907625237690847140566322523616, −9.748334141919165252759584402859, −9.032203529854875742104128708192, −8.843608320647336394721236184677, −8.314892281876961100167798868439, −8.125337182673025316293840995183, −7.18430078508535521910330045958, −6.74511239064606168880527937350, −6.00536895441430038222988247375, −5.93507421821593643474007710963, −5.01497589773578907578475957798, −4.84630458177823621633460550463, −4.15988395531285717430007130557, −3.85068137041221218914294056354, −3.33485417217454320487390575690, −2.73434850078262095353672390450, −1.54449135029953767081543110981, −1.16751557352795145810139194714, 0, 0,
1.16751557352795145810139194714, 1.54449135029953767081543110981, 2.73434850078262095353672390450, 3.33485417217454320487390575690, 3.85068137041221218914294056354, 4.15988395531285717430007130557, 4.84630458177823621633460550463, 5.01497589773578907578475957798, 5.93507421821593643474007710963, 6.00536895441430038222988247375, 6.74511239064606168880527937350, 7.18430078508535521910330045958, 8.125337182673025316293840995183, 8.314892281876961100167798868439, 8.843608320647336394721236184677, 9.032203529854875742104128708192, 9.748334141919165252759584402859, 9.907625237690847140566322523616
