L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + 0.267i·5-s + (0.633 − 0.366i)7-s − 0.999i·8-s + (0.133 − 0.232i)10-s + (4.09 + 2.36i)11-s + (2.59 − 2.5i)13-s − 0.732·14-s + (−0.5 + 0.866i)16-s + (1.13 + 1.96i)17-s + (−1.09 + 0.633i)19-s + (−0.232 + 0.133i)20-s + (−2.36 − 4.09i)22-s + (3.09 − 5.36i)23-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + 0.119i·5-s + (0.239 − 0.138i)7-s − 0.353i·8-s + (0.0423 − 0.0733i)10-s + (1.23 + 0.713i)11-s + (0.720 − 0.693i)13-s − 0.195·14-s + (−0.125 + 0.216i)16-s + (0.275 + 0.476i)17-s + (−0.251 + 0.145i)19-s + (−0.0518 + 0.0299i)20-s + (−0.504 − 0.873i)22-s + (0.645 − 1.11i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01395 - 0.136797i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01395 - 0.136797i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-2.59 + 2.5i)T \) |
good | 5 | \( 1 - 0.267iT - 5T^{2} \) |
| 7 | \( 1 + (-0.633 + 0.366i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.09 - 2.36i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.13 - 1.96i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.09 - 0.633i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.09 + 5.36i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.23 + 2.13i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 5.46iT - 31T^{2} \) |
| 37 | \( 1 + (9.06 + 5.23i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (9.86 + 5.69i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.83 - 6.63i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 8.19iT - 47T^{2} \) |
| 53 | \( 1 + 0.464T + 53T^{2} \) |
| 59 | \( 1 + (6.92 - 4i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.598 + 1.03i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.63 + 5.56i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.09 + 0.633i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 9.73iT - 73T^{2} \) |
| 79 | \( 1 + 9.46T + 79T^{2} \) |
| 83 | \( 1 - 10.1iT - 83T^{2} \) |
| 89 | \( 1 + (2.19 + 1.26i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.19 + 3i)T + (48.5 - 84.0i)T^{2} \) |
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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
−12.17421358038848320259390721500, −10.91183557192232317902172983791, −10.39344623158946352913326105992, −9.141268244040168162037612198361, −8.426756166077192505814119931566, −7.20235412890674974704679129409, −6.25315135804621490756124117338, −4.57272560014705642081958134468, −3.24691836981778020182632505872, −1.45288932891623907853916003450,
1.42157652293675106491248603323, 3.50236290955058264205424282769, 5.07208780874797683069112869386, 6.32760962051973118848893813388, 7.15661610055568063671590149645, 8.603075723631020551871269840809, 8.986267114335924353467551216645, 10.17190163259953431505856969127, 11.37355399595108756212105255751, 11.81122528002255499482166055748