L(s) = 1 | − 19.6·3-s + 46.7·5-s + 49·7-s + 143.·9-s − 666.·11-s + 650.·13-s − 918.·15-s + 1.18e3·17-s + 1.56e3·19-s − 962.·21-s − 1.10e3·23-s − 939.·25-s + 1.96e3·27-s − 2.39e3·29-s − 2.04e3·31-s + 1.30e4·33-s + 2.29e3·35-s − 1.07e3·37-s − 1.27e4·39-s + 1.09e3·41-s − 1.65e4·43-s + 6.68e3·45-s − 8.29e3·47-s + 2.40e3·49-s − 2.33e4·51-s − 5.51e3·53-s − 3.11e4·55-s + ⋯ |
L(s) = 1 | − 1.26·3-s + 0.836·5-s + 0.377·7-s + 0.588·9-s − 1.65·11-s + 1.06·13-s − 1.05·15-s + 0.996·17-s + 0.994·19-s − 0.476·21-s − 0.433·23-s − 0.300·25-s + 0.518·27-s − 0.529·29-s − 0.382·31-s + 2.09·33-s + 0.316·35-s − 0.129·37-s − 1.34·39-s + 0.102·41-s − 1.36·43-s + 0.492·45-s − 0.547·47-s + 0.142·49-s − 1.25·51-s − 0.269·53-s − 1.38·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.385819565\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.385819565\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - 49T \) |
good | 3 | \( 1 + 19.6T + 243T^{2} \) |
| 5 | \( 1 - 46.7T + 3.12e3T^{2} \) |
| 11 | \( 1 + 666.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 650.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.18e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.56e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.10e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.07e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.09e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.65e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.29e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.51e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.97e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.06e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 675.T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.25e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.29e4T + 8.58e9T^{2} \) |
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
−10.37587066907763670113096966240, −9.749993719601740940690952078028, −8.375778896660566978483976830225, −7.49225709612504135845680461546, −6.20835041791068894624075030159, −5.52460876808262897673579306679, −5.02820586963105823828760650706, −3.34334747292362858618442178671, −1.87212605384443258163959413052, −0.65259276939541625870991525234,
0.65259276939541625870991525234, 1.87212605384443258163959413052, 3.34334747292362858618442178671, 5.02820586963105823828760650706, 5.52460876808262897673579306679, 6.20835041791068894624075030159, 7.49225709612504135845680461546, 8.375778896660566978483976830225, 9.749993719601740940690952078028, 10.37587066907763670113096966240