| L(s) = 1 | − 4-s + 16-s − 4·19-s + 12·29-s + 16·31-s − 12·41-s − 49-s + 24·59-s + 16·61-s − 64-s − 12·71-s + 4·76-s + 32·79-s + 12·89-s − 12·101-s + 20·109-s − 12·116-s − 22·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1/4·16-s − 0.917·19-s + 2.22·29-s + 2.87·31-s − 1.87·41-s − 1/7·49-s + 3.12·59-s + 2.04·61-s − 1/8·64-s − 1.42·71-s + 0.458·76-s + 3.60·79-s + 1.27·89-s − 1.19·101-s + 1.91·109-s − 1.11·116-s − 2·121-s − 1.43·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.716287376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.716287376\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.538725681242857869666908292632, −8.482748299997655803667908699491, −8.200121323913554069772071555711, −8.122393540218572032822276816006, −7.28002282381830107871109533855, −6.90582404766489243037836502391, −6.52300765544816658970992038916, −6.44244083100468576861231443861, −5.88210687213051791097291490817, −5.26336648189400356114605460107, −5.06083094473874519121372982960, −4.54944079256386358730642439459, −4.33758145465434388558654496862, −3.79068177135856846222756450670, −3.29071773038933164007453995990, −2.84891059370008941722530675145, −2.31612230040998051861296662072, −1.89248978017999300971331797570, −0.830246319784123610551317756921, −0.73863750808720447045536713662,
0.73863750808720447045536713662, 0.830246319784123610551317756921, 1.89248978017999300971331797570, 2.31612230040998051861296662072, 2.84891059370008941722530675145, 3.29071773038933164007453995990, 3.79068177135856846222756450670, 4.33758145465434388558654496862, 4.54944079256386358730642439459, 5.06083094473874519121372982960, 5.26336648189400356114605460107, 5.88210687213051791097291490817, 6.44244083100468576861231443861, 6.52300765544816658970992038916, 6.90582404766489243037836502391, 7.28002282381830107871109533855, 8.122393540218572032822276816006, 8.200121323913554069772071555711, 8.482748299997655803667908699491, 8.538725681242857869666908292632
