| L(s) = 1 | − 9·3-s + 16·4-s + 81·9-s − 144·12-s − 191·13-s + 256·16-s + 601·19-s + 625·25-s − 729·27-s + 1.75e3·31-s + 1.29e3·36-s + 2.59e3·37-s + 1.71e3·39-s + 23·43-s − 2.30e3·48-s − 3.05e3·52-s − 5.40e3·57-s + 1.96e3·61-s + 4.09e3·64-s − 8.80e3·67-s + 1.24e3·73-s − 5.62e3·75-s + 9.61e3·76-s − 1.23e4·79-s + 6.56e3·81-s − 1.57e4·93-s + 1.88e4·97-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 9-s − 12-s − 1.13·13-s + 16-s + 1.66·19-s + 25-s − 27-s + 1.82·31-s + 36-s + 1.89·37-s + 1.13·39-s + 0.0124·43-s − 48-s − 1.13·52-s − 1.66·57-s + 0.528·61-s + 64-s − 1.96·67-s + 0.234·73-s − 75-s + 1.66·76-s − 1.98·79-s + 81-s − 1.82·93-s + 1.99·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.698192176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.698192176\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{2} T \) |
| 7 | \( 1 \) |
| good | 2 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 5 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( 1 + 191 T + p^{4} T^{2} \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( 1 - 601 T + p^{4} T^{2} \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 31 | \( 1 - 1753 T + p^{4} T^{2} \) |
| 37 | \( 1 - 2591 T + p^{4} T^{2} \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( 1 - 23 T + p^{4} T^{2} \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( 1 - 1966 T + p^{4} T^{2} \) |
| 67 | \( 1 + 8809 T + p^{4} T^{2} \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 - 1249 T + p^{4} T^{2} \) |
| 79 | \( 1 + 12361 T + p^{4} T^{2} \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( 1 - 18814 T + p^{4} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.00532718758095546555511932915, −11.57359376679852890377593645115, −10.43883791181472922949539739788, −9.648859235791804409407255140385, −7.75118013188515195453699447002, −6.93761269921816801224797773750, −5.86779749929220422214041966527, −4.72168877936399098743693119823, −2.79777638642665487287250953226, −1.04423461037355702322409602619,
1.04423461037355702322409602619, 2.79777638642665487287250953226, 4.72168877936399098743693119823, 5.86779749929220422214041966527, 6.93761269921816801224797773750, 7.75118013188515195453699447002, 9.648859235791804409407255140385, 10.43883791181472922949539739788, 11.57359376679852890377593645115, 12.00532718758095546555511932915
