L(s) = 1 | − 3-s + 9-s + 4·13-s − 8·23-s − 2·25-s − 27-s − 4·37-s − 4·39-s − 8·47-s + 6·49-s + 8·59-s − 4·61-s + 8·69-s − 8·71-s − 4·73-s + 2·75-s + 81-s − 16·83-s − 12·97-s − 8·107-s + 20·109-s + 4·111-s + 4·117-s − 22·121-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.10·13-s − 1.66·23-s − 2/5·25-s − 0.192·27-s − 0.657·37-s − 0.640·39-s − 1.16·47-s + 6/7·49-s + 1.04·59-s − 0.512·61-s + 0.963·69-s − 0.949·71-s − 0.468·73-s + 0.230·75-s + 1/9·81-s − 1.75·83-s − 1.21·97-s − 0.773·107-s + 1.91·109-s + 0.379·111-s + 0.369·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.428204091467166702776829331885, −7.908364750178132282218784842729, −7.45179059701174234991084085500, −6.92615445512589699035486591670, −6.42186070370104700260184890550, −6.01833448398747410267957503004, −5.63157460903723921469278047199, −5.15320725629932979230395236706, −4.40858075714082169311806488267, −4.00765993071215532880405262093, −3.53754748554363847039164208978, −2.75965540841893175141685581236, −1.91941394435507967335965249650, −1.28067062182510752966829554192, 0,
1.28067062182510752966829554192, 1.91941394435507967335965249650, 2.75965540841893175141685581236, 3.53754748554363847039164208978, 4.00765993071215532880405262093, 4.40858075714082169311806488267, 5.15320725629932979230395236706, 5.63157460903723921469278047199, 6.01833448398747410267957503004, 6.42186070370104700260184890550, 6.92615445512589699035486591670, 7.45179059701174234991084085500, 7.908364750178132282218784842729, 8.428204091467166702776829331885