| L(s) = 1 | + 3·3-s − 5-s + 4·7-s + 3·9-s − 11-s − 4·13-s − 3·15-s − 3·17-s + 5·19-s + 12·21-s + 3·23-s + 5·25-s + 12·29-s + 31-s − 3·33-s − 4·35-s − 5·37-s − 12·39-s − 20·41-s + 8·43-s − 3·45-s − 47-s + 9·49-s − 9·51-s − 9·53-s + 55-s + 15·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s + 1.51·7-s + 9-s − 0.301·11-s − 1.10·13-s − 0.774·15-s − 0.727·17-s + 1.14·19-s + 2.61·21-s + 0.625·23-s + 25-s + 2.22·29-s + 0.179·31-s − 0.522·33-s − 0.676·35-s − 0.821·37-s − 1.92·39-s − 3.12·41-s + 1.21·43-s − 0.447·45-s − 0.145·47-s + 9/7·49-s − 1.26·51-s − 1.23·53-s + 0.134·55-s + 1.98·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.238715888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.238715888\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−11.18143546427841463818942759146, −10.92763573543515004069109488814, −10.42784991413781463946568626589, −9.797425823616321587668115275112, −9.502098050547633598934452136032, −8.793044937181622240119597585892, −8.581922835133778278259751725937, −8.080453914350116039418667028901, −8.031987535371007416867774387683, −7.33532063233157414062992868651, −6.86698067804866699275782873062, −6.46639484243756491199304551933, −5.11904549307203993490682241331, −5.02565532107304960916713457428, −4.75245387324220719512885046039, −3.68900193758903036522364230515, −3.30848895295155402462994362908, −2.52450582703725952683946258894, −2.25056367087603332020954781162, −1.13387679500666644814933230384,
1.13387679500666644814933230384, 2.25056367087603332020954781162, 2.52450582703725952683946258894, 3.30848895295155402462994362908, 3.68900193758903036522364230515, 4.75245387324220719512885046039, 5.02565532107304960916713457428, 5.11904549307203993490682241331, 6.46639484243756491199304551933, 6.86698067804866699275782873062, 7.33532063233157414062992868651, 8.031987535371007416867774387683, 8.080453914350116039418667028901, 8.581922835133778278259751725937, 8.793044937181622240119597585892, 9.502098050547633598934452136032, 9.797425823616321587668115275112, 10.42784991413781463946568626589, 10.92763573543515004069109488814, 11.18143546427841463818942759146
