| 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
| \(L(1)\) |
\(\approx\) |
\(3.258604536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.258604536\) |
| \(L(\frac{3}{2})\) |
|
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
−9.052016631423950708058146424336, −8.511897890900031397586163017766, −8.269697574941950769499911919014, −7.62401279918674081669303650657, −7.33441252050001999945408146987, −6.93912965052025744754997615192, −6.55630753306782811995993570802, −6.48444055013073773531210707930, −5.79944925600103075498291299965, −5.38424556257030379627518165271, −5.06451357858188723652999491533, −4.50922004302943122149772704130, −4.21826392876047855227623197106, −3.78401417149802955330425078583, −3.32993007022582528094463001743, −2.82483649072711603000692070442, −2.39109042500610377097435741136, −1.44133962431065759803786858619, −1.40899889727374605756137633540, −0.60036600929581647792231505936,
0.60036600929581647792231505936, 1.40899889727374605756137633540, 1.44133962431065759803786858619, 2.39109042500610377097435741136, 2.82483649072711603000692070442, 3.32993007022582528094463001743, 3.78401417149802955330425078583, 4.21826392876047855227623197106, 4.50922004302943122149772704130, 5.06451357858188723652999491533, 5.38424556257030379627518165271, 5.79944925600103075498291299965, 6.48444055013073773531210707930, 6.55630753306782811995993570802, 6.93912965052025744754997615192, 7.33441252050001999945408146987, 7.62401279918674081669303650657, 8.269697574941950769499911919014, 8.511897890900031397586163017766, 9.052016631423950708058146424336
