L(s) = 1 | − 2.74i·2-s − 1.73i·3-s − 3.55·4-s − 0.569i·5-s − 4.76·6-s − 7.88i·7-s − 1.21i·8-s − 2.99·9-s − 1.56·10-s + 7.66i·11-s + 6.16i·12-s − 2.48i·13-s − 21.6·14-s − 0.987·15-s − 17.5·16-s − 20.2·17-s + ⋯ |
L(s) = 1 | − 1.37i·2-s − 0.577i·3-s − 0.889·4-s − 0.113i·5-s − 0.793·6-s − 1.12i·7-s − 0.152i·8-s − 0.333·9-s − 0.156·10-s + 0.696i·11-s + 0.513i·12-s − 0.191i·13-s − 1.54·14-s − 0.0658·15-s − 1.09·16-s − 1.19·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 - 0.424i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.282177 + 1.26583i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.282177 + 1.26583i\) |
\(L(2)\) |
|
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 | \( 1 + (28.4 - 60.6i)T \) |
good | 2 | \( 1 + 2.74iT - 4T^{2} \) |
| 5 | \( 1 + 0.569iT - 25T^{2} \) |
| 7 | \( 1 + 7.88iT - 49T^{2} \) |
| 11 | \( 1 - 7.66iT - 121T^{2} \) |
| 13 | \( 1 + 2.48iT - 169T^{2} \) |
| 17 | \( 1 + 20.2T + 289T^{2} \) |
| 19 | \( 1 - 13.9T + 361T^{2} \) |
| 23 | \( 1 - 10.7T + 529T^{2} \) |
| 29 | \( 1 + 27.0T + 841T^{2} \) |
| 31 | \( 1 + 2.26iT - 961T^{2} \) |
| 37 | \( 1 - 53.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 16.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 44.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 2.48T + 2.20e3T^{2} \) |
| 53 | \( 1 + 86.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 16.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 97.3iT - 3.72e3T^{2} \) |
| 71 | \( 1 - 11.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 58.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 0.870iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 47.5T + 6.88e3T^{2} \) |
| 89 | \( 1 - 47.1T + 7.92e3T^{2} \) |
| 97 | \( 1 + 21.2iT - 9.40e3T^{2} \) |
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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
−11.53659395844307253119611366631, −10.89735337599431889839864304031, −9.965216539588767429734761111051, −8.987348486714727967610495594647, −7.49997241540730643953299732969, −6.69130878271539170239714969420, −4.78930016499217843686760891123, −3.60274670825700897362581839345, −2.15295074162342144042999759203, −0.74793172230511312905632938547,
2.75466911083400343695325812702, 4.60883193627550086662589343533, 5.64285385844942714626671578016, 6.43323516120172338089246159355, 7.69235551562060618770200344540, 8.820200414738811173104643157994, 9.262277738888531827153885957877, 10.92974689275094931278122344623, 11.63379942841564068438902385913, 13.04627260371936153339023650312