L(s) = 1 | + 81·3-s + 239·7-s + 6.56e3·9-s − 2.06e4·13-s + 1.00e5·19-s + 1.93e4·21-s + 5.31e5·27-s + 5.83e5·31-s + 5.03e5·37-s − 1.67e6·39-s + 3.34e6·43-s − 5.70e6·49-s + 8.14e6·57-s + 2.41e7·61-s + 1.56e6·63-s + 3.72e7·67-s + 1.61e7·73-s − 1.88e7·79-s + 4.30e7·81-s − 4.93e6·91-s + 4.72e7·93-s − 9.47e7·97-s + 4.44e7·103-s + 6.81e7·109-s + 4.07e7·111-s − 1.35e8·117-s + ⋯ |
L(s) = 1 | + 3-s + 0.0995·7-s + 9-s − 0.722·13-s + 0.771·19-s + 0.0995·21-s + 27-s + 0.631·31-s + 0.268·37-s − 0.722·39-s + 0.978·43-s − 0.990·49-s + 0.771·57-s + 1.74·61-s + 0.0995·63-s + 1.85·67-s + 0.569·73-s − 0.484·79-s + 81-s − 0.0719·91-s + 0.631·93-s − 1.07·97-s + 0.394·103-s + 0.482·109-s + 0.268·111-s − 0.722·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(3.560210414\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.560210414\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{4} T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 239 T + p^{8} T^{2} \) |
| 11 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 13 | \( 1 + 20641 T + p^{8} T^{2} \) |
| 17 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 19 | \( 1 - 100559 T + p^{8} T^{2} \) |
| 23 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 29 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 31 | \( 1 - 583439 T + p^{8} T^{2} \) |
| 37 | \( 1 - 503522 T + p^{8} T^{2} \) |
| 41 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 43 | \( 1 - 3344879 T + p^{8} T^{2} \) |
| 47 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 53 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 59 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 61 | \( 1 - 24133919 T + p^{8} T^{2} \) |
| 67 | \( 1 - 37296239 T + p^{8} T^{2} \) |
| 71 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 73 | \( 1 - 16169282 T + p^{8} T^{2} \) |
| 79 | \( 1 + 18887038 T + p^{8} T^{2} \) |
| 83 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 89 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 97 | \( 1 + 94775521 T + p^{8} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.04505463951778579049226672843, −9.457812558833705499932708867167, −8.401761480301838353399774150909, −7.60860803114809691525833758219, −6.70124771773656057777228184492, −5.25308503247455011493072546598, −4.17684434029253983358912484936, −3.06088957976751868315680744958, −2.10519632326143796356731070066, −0.845530809961226658113449772068,
0.845530809961226658113449772068, 2.10519632326143796356731070066, 3.06088957976751868315680744958, 4.17684434029253983358912484936, 5.25308503247455011493072546598, 6.70124771773656057777228184492, 7.60860803114809691525833758219, 8.401761480301838353399774150909, 9.457812558833705499932708867167, 10.04505463951778579049226672843