L(s) = 1 | + 4·3-s + 3·4-s + 6·9-s + 12·12-s − 4·13-s + 5·16-s + 12·23-s − 4·27-s − 12·29-s + 18·36-s − 16·39-s − 12·43-s + 20·48-s + 14·49-s − 12·52-s − 24·53-s + 12·61-s + 3·64-s + 48·69-s − 37·81-s − 48·87-s + 36·92-s + 12·101-s + 12·103-s + 12·107-s − 12·108-s − 36·116-s + ⋯ |
L(s) = 1 | + 2.30·3-s + 3/2·4-s + 2·9-s + 3.46·12-s − 1.10·13-s + 5/4·16-s + 2.50·23-s − 0.769·27-s − 2.22·29-s + 3·36-s − 2.56·39-s − 1.82·43-s + 2.88·48-s + 2·49-s − 1.66·52-s − 3.29·53-s + 1.53·61-s + 3/8·64-s + 5.77·69-s − 4.11·81-s − 5.14·87-s + 3.75·92-s + 1.19·101-s + 1.18·103-s + 1.16·107-s − 1.15·108-s − 3.34·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.288361690\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.288361690\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−11.51097974961019453693761398593, −11.48335480489018205382092358843, −11.04126496168835771267326474181, −10.38546052576691879232312697168, −9.793977656800040321730176468921, −9.432934503057887530189474089897, −8.886620010724159412422674149030, −8.723785406429474636325079626613, −7.83657285223277931890250951501, −7.72138239672211531780943250613, −7.14920297447368846528732085139, −6.91087513343003390718548320993, −6.09016604512008503144563533649, −5.39606012788722682752404915554, −4.79770767370664352185513698856, −3.73961609501620932225141553181, −3.20114426675046430668429134682, −2.90609290108933831304524676338, −2.19968394585130162988212608852, −1.77144920368671874843295631541,
1.77144920368671874843295631541, 2.19968394585130162988212608852, 2.90609290108933831304524676338, 3.20114426675046430668429134682, 3.73961609501620932225141553181, 4.79770767370664352185513698856, 5.39606012788722682752404915554, 6.09016604512008503144563533649, 6.91087513343003390718548320993, 7.14920297447368846528732085139, 7.72138239672211531780943250613, 7.83657285223277931890250951501, 8.723785406429474636325079626613, 8.886620010724159412422674149030, 9.432934503057887530189474089897, 9.793977656800040321730176468921, 10.38546052576691879232312697168, 11.04126496168835771267326474181, 11.48335480489018205382092358843, 11.51097974961019453693761398593