| L(s) = 1 | + 2·3-s − 2·5-s + 6·7-s + 2·9-s − 6·13-s − 4·15-s + 2·17-s + 8·19-s + 12·21-s + 2·23-s − 25-s + 6·27-s − 12·35-s + 2·37-s − 12·39-s − 20·41-s + 10·43-s − 4·45-s + 6·47-s + 18·49-s + 4·51-s + 10·53-s + 16·57-s − 24·59-s − 4·61-s + 12·63-s + 12·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 2.26·7-s + 2/3·9-s − 1.66·13-s − 1.03·15-s + 0.485·17-s + 1.83·19-s + 2.61·21-s + 0.417·23-s − 1/5·25-s + 1.15·27-s − 2.02·35-s + 0.328·37-s − 1.92·39-s − 3.12·41-s + 1.52·43-s − 0.596·45-s + 0.875·47-s + 18/7·49-s + 0.560·51-s + 1.37·53-s + 2.11·57-s − 3.12·59-s − 0.512·61-s + 1.51·63-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.486942588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.486942588\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−11.81591366705771694113443395954, −11.60867118319147795173080593192, −10.91406864582119726877925109974, −10.57734156767100509843644300402, −9.863288933263972862940294754371, −9.574229728358341599138416755870, −8.721526712404733872751331346150, −8.664646218729736647431524457543, −7.926287177670478020845990093964, −7.65693164585946708203716146088, −7.41312387154114080138329532482, −6.96600513175471902469013514923, −5.76657328954933454743612435381, −5.16562771064088594595218800447, −4.72433740835131608138449855835, −4.39210322439505830010682398136, −3.42519377028204504433773615563, −2.92838589410769449520653675639, −2.09253552181524448462198734667, −1.25787883574846595639059045570,
1.25787883574846595639059045570, 2.09253552181524448462198734667, 2.92838589410769449520653675639, 3.42519377028204504433773615563, 4.39210322439505830010682398136, 4.72433740835131608138449855835, 5.16562771064088594595218800447, 5.76657328954933454743612435381, 6.96600513175471902469013514923, 7.41312387154114080138329532482, 7.65693164585946708203716146088, 7.926287177670478020845990093964, 8.664646218729736647431524457543, 8.721526712404733872751331346150, 9.574229728358341599138416755870, 9.863288933263972862940294754371, 10.57734156767100509843644300402, 10.91406864582119726877925109974, 11.60867118319147795173080593192, 11.81591366705771694113443395954
