| L(s) = 1 | + 2·9-s − 6·11-s + 4·19-s − 18·29-s − 16·31-s − 12·41-s − 49-s − 20·61-s − 30·71-s − 14·79-s − 5·81-s + 24·89-s − 12·99-s + 36·101-s + 14·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 8·171-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 1.80·11-s + 0.917·19-s − 3.34·29-s − 2.87·31-s − 1.87·41-s − 1/7·49-s − 2.56·61-s − 3.56·71-s − 1.57·79-s − 5/9·81-s + 2.54·89-s − 1.20·99-s + 3.58·101-s + 1.34·109-s + 5/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.611·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.913383109175226540092754157803, −8.058823179046715229263175723914, −7.60791193984134380716649530272, −7.46181941086381589019593464721, −7.41844493321202473710536866686, −6.91959438946699318308594335924, −6.18628571674854184302972098868, −5.79250126945184046131368402227, −5.49416752198638095866245041426, −5.25885429901591757925485511255, −4.70804525865876651982136532390, −4.35608370375259282853289871296, −3.65526627989818650491721248992, −3.31419399824660808740319286938, −3.10430847783344829519098190044, −2.16878031623773248339956403444, −1.84758571978563807409937852785, −1.44325447619435435747505969879, 0, 0,
1.44325447619435435747505969879, 1.84758571978563807409937852785, 2.16878031623773248339956403444, 3.10430847783344829519098190044, 3.31419399824660808740319286938, 3.65526627989818650491721248992, 4.35608370375259282853289871296, 4.70804525865876651982136532390, 5.25885429901591757925485511255, 5.49416752198638095866245041426, 5.79250126945184046131368402227, 6.18628571674854184302972098868, 6.91959438946699318308594335924, 7.41844493321202473710536866686, 7.46181941086381589019593464721, 7.60791193984134380716649530272, 8.058823179046715229263175723914, 8.913383109175226540092754157803
