| L(s) = 1 | + 2·9-s − 4·11-s − 4·19-s − 12·29-s − 12·41-s − 2·49-s − 28·59-s − 4·61-s − 24·71-s − 16·79-s − 5·81-s + 4·89-s − 8·99-s + 12·101-s − 12·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s − 8·171-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 1.20·11-s − 0.917·19-s − 2.22·29-s − 1.87·41-s − 2/7·49-s − 3.64·59-s − 0.512·61-s − 2.84·71-s − 1.80·79-s − 5/9·81-s + 0.423·89-s − 0.804·99-s + 1.19·101-s − 1.14·109-s − 0.909·121-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 + 1.69·169-s − 0.611·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.334441302051277550921406765719, −8.114914068764887856369628062794, −7.67455813663108951542839391754, −7.48291971878980060725313575119, −6.90543675263443085225841505650, −6.82976290048234225091642455929, −6.02551992111356855047994170118, −5.90041518263471523355653791751, −5.50003191517293580533629521356, −4.97321831644272694877932614841, −4.45771319679722568751419695004, −4.42839328559680894651992689943, −3.72681319535339418447199570017, −3.16317372166788583079257348997, −2.97935431906957312171637388271, −2.23103731694955860958888037925, −1.68107224980980954555286641825, −1.46038698318094762408882154618, 0, 0,
1.46038698318094762408882154618, 1.68107224980980954555286641825, 2.23103731694955860958888037925, 2.97935431906957312171637388271, 3.16317372166788583079257348997, 3.72681319535339418447199570017, 4.42839328559680894651992689943, 4.45771319679722568751419695004, 4.97321831644272694877932614841, 5.50003191517293580533629521356, 5.90041518263471523355653791751, 6.02551992111356855047994170118, 6.82976290048234225091642455929, 6.90543675263443085225841505650, 7.48291971878980060725313575119, 7.67455813663108951542839391754, 8.114914068764887856369628062794, 8.334441302051277550921406765719
