L(s) = 1 | + 1.70i·2-s + 1.73i·3-s + 1.09·4-s − 1.67i·5-s − 2.95·6-s − 0.231i·7-s + 8.68i·8-s − 2.99·9-s + 2.85·10-s + 17.0i·11-s + 1.90i·12-s + 6.17i·13-s + 0.394·14-s + 2.89·15-s − 10.4·16-s + 15.1·17-s + ⋯ |
L(s) = 1 | + 0.851i·2-s + 0.577i·3-s + 0.274·4-s − 0.334i·5-s − 0.491·6-s − 0.0330i·7-s + 1.08i·8-s − 0.333·9-s + 0.285·10-s + 1.55i·11-s + 0.158i·12-s + 0.475i·13-s + 0.0281·14-s + 0.193·15-s − 0.650·16-s + 0.889·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.680 - 0.732i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.654309 + 1.50089i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.654309 + 1.50089i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73iT \) |
| 67 | \( 1 + (-49.0 + 45.6i)T \) |
good | 2 | \( 1 - 1.70iT - 4T^{2} \) |
| 5 | \( 1 + 1.67iT - 25T^{2} \) |
| 7 | \( 1 + 0.231iT - 49T^{2} \) |
| 11 | \( 1 - 17.0iT - 121T^{2} \) |
| 13 | \( 1 - 6.17iT - 169T^{2} \) |
| 17 | \( 1 - 15.1T + 289T^{2} \) |
| 19 | \( 1 + 6.48T + 361T^{2} \) |
| 23 | \( 1 + 37.1T + 529T^{2} \) |
| 29 | \( 1 - 24.7T + 841T^{2} \) |
| 31 | \( 1 + 48.8iT - 961T^{2} \) |
| 37 | \( 1 + 20.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 75.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 60.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 71.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 88.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 7.13T + 3.48e3T^{2} \) |
| 61 | \( 1 - 41.0iT - 3.72e3T^{2} \) |
| 71 | \( 1 + 0.114T + 5.04e3T^{2} \) |
| 73 | \( 1 - 58.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 24.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 3.41T + 6.88e3T^{2} \) |
| 89 | \( 1 - 147.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 50.6iT - 9.40e3T^{2} \) |
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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
−12.39864168339165510168539074515, −11.74336772775278891652044038758, −10.43863460607036687718619747398, −9.627053273845613617791672949339, −8.401750868054159290079758418770, −7.46968534079392438035670889904, −6.40914394190037991209977847012, −5.24943738416078013045712664316, −4.18658691834079124057393575947, −2.20423514818176257595938542348,
0.972994011982118479595565673676, 2.63688298846085149103309797286, 3.61521199329934502848808837062, 5.69116197976866619867642976065, 6.63585522596200241456623782868, 7.82107851237363993891193163831, 8.884428895212389284346455868893, 10.42029913439736288373291204085, 10.78222543895687934268246604635, 12.03055551714879846194971455343