| L(s) = 1 | + 2·7-s − 6·9-s − 12·17-s − 6·25-s − 16·31-s + 4·41-s + 16·47-s + 3·49-s − 12·63-s + 16·71-s + 20·73-s − 32·79-s + 27·81-s − 12·89-s − 12·97-s + 32·103-s + 4·113-s − 24·119-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 72·153-s + 157-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 2·9-s − 2.91·17-s − 6/5·25-s − 2.87·31-s + 0.624·41-s + 2.33·47-s + 3/7·49-s − 1.51·63-s + 1.89·71-s + 2.34·73-s − 3.60·79-s + 3·81-s − 1.27·89-s − 1.21·97-s + 3.15·103-s + 0.376·113-s − 2.20·119-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 5.82·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−9.688287404638284990728895668207, −9.117894034638208548514808203742, −8.749304120934004278681759852831, −8.560607806232820418165457845354, −7.79292083571821938290976650763, −7.26382021666693896035142333427, −6.67949194679031572918581545842, −5.82613210142107824134972732669, −5.69394971603405508863657314834, −4.92225264716551837532790929786, −4.18465932715971087267449515138, −3.60173413237973468362473708205, −2.40145156924384381758797773878, −2.14556791802447766709825303506, 0,
2.14556791802447766709825303506, 2.40145156924384381758797773878, 3.60173413237973468362473708205, 4.18465932715971087267449515138, 4.92225264716551837532790929786, 5.69394971603405508863657314834, 5.82613210142107824134972732669, 6.67949194679031572918581545842, 7.26382021666693896035142333427, 7.79292083571821938290976650763, 8.560607806232820418165457845354, 8.749304120934004278681759852831, 9.117894034638208548514808203742, 9.688287404638284990728895668207
