| L(s) = 1 | − 5·7-s − 3·9-s − 3·13-s − 4·16-s − 3·19-s + 7·25-s + 7·31-s + 37-s − 5·43-s + 18·49-s + 21·61-s + 15·63-s + 14·67-s + 24·73-s − 8·79-s + 9·81-s + 15·91-s − 18·97-s − 21·103-s + 4·109-s + 20·112-s + 9·117-s + 2·121-s + 127-s + 131-s + 15·133-s + 137-s + ⋯ |
| L(s) = 1 | − 1.88·7-s − 9-s − 0.832·13-s − 16-s − 0.688·19-s + 7/5·25-s + 1.25·31-s + 0.164·37-s − 0.762·43-s + 18/7·49-s + 2.68·61-s + 1.88·63-s + 1.71·67-s + 2.80·73-s − 0.900·79-s + 81-s + 1.57·91-s − 1.82·97-s − 2.06·103-s + 0.383·109-s + 1.88·112-s + 0.832·117-s + 2/11·121-s + 0.0887·127-s + 0.0873·131-s + 1.30·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900081 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900081 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 157 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 22 T + p T^{2} ) \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 91 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.222538500393866316513067776036, −7.31752229824145705426293579893, −6.89011305781361004363000050077, −6.67478865509709061098247476582, −6.33732212366292724469218962420, −5.81303055953477417093404684482, −5.08585764403305772783381194354, −4.99590881680675287407385676664, −4.08522580190036348341434383379, −3.71610470373537955702604622160, −3.06228783641856972416051651661, −2.45873690170705660156081996400, −2.40351183741868248506337105269, −0.834333049329446256581190107861, 0,
0.834333049329446256581190107861, 2.40351183741868248506337105269, 2.45873690170705660156081996400, 3.06228783641856972416051651661, 3.71610470373537955702604622160, 4.08522580190036348341434383379, 4.99590881680675287407385676664, 5.08585764403305772783381194354, 5.81303055953477417093404684482, 6.33732212366292724469218962420, 6.67478865509709061098247476582, 6.89011305781361004363000050077, 7.31752229824145705426293579893, 8.222538500393866316513067776036
