L(s) = 1 | − 2.60i·2-s + (2.93 + 0.635i)3-s − 2.80·4-s − 2.23i·5-s + (1.65 − 7.65i)6-s − 2.64·7-s − 3.10i·8-s + (8.19 + 3.72i)9-s − 5.83·10-s − 7.66i·11-s + (−8.23 − 1.78i)12-s + 12.2·13-s + 6.90i·14-s + (1.42 − 6.55i)15-s − 19.3·16-s + 28.0i·17-s + ⋯ |
L(s) = 1 | − 1.30i·2-s + (0.977 + 0.211i)3-s − 0.702·4-s − 0.447i·5-s + (0.276 − 1.27i)6-s − 0.377·7-s − 0.388i·8-s + (0.910 + 0.414i)9-s − 0.583·10-s − 0.697i·11-s + (−0.686 − 0.148i)12-s + 0.944·13-s + 0.493i·14-s + (0.0947 − 0.437i)15-s − 1.20·16-s + 1.64i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 + 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.12919 - 1.40017i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12919 - 1.40017i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.93 - 0.635i)T \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + 2.64T \) |
good | 2 | \( 1 + 2.60iT - 4T^{2} \) |
| 11 | \( 1 + 7.66iT - 121T^{2} \) |
| 13 | \( 1 - 12.2T + 169T^{2} \) |
| 17 | \( 1 - 28.0iT - 289T^{2} \) |
| 19 | \( 1 + 22.7T + 361T^{2} \) |
| 23 | \( 1 - 18.7iT - 529T^{2} \) |
| 29 | \( 1 - 13.5iT - 841T^{2} \) |
| 31 | \( 1 - 59.4T + 961T^{2} \) |
| 37 | \( 1 + 28.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 11.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 4.45T + 1.84e3T^{2} \) |
| 47 | \( 1 - 6.20iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 38.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 106. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 55.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 87.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 29.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 0.585T + 5.32e3T^{2} \) |
| 79 | \( 1 - 7.71T + 6.24e3T^{2} \) |
| 83 | \( 1 - 62.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 150. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 36.3T + 9.40e3T^{2} \) |
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
−13.12673980226805161860759942225, −12.31186076157034855363783337617, −10.91516841487211555529210364027, −10.21059619945693218953499074826, −9.005131736196550698938612497367, −8.244222115107324390599284643015, −6.35448401879304505134313738527, −4.19112791801685356874368704403, −3.23498744516451651187119070788, −1.60422189788766595350113957848,
2.62319591273401683787312871805, 4.48940341204050617983597298074, 6.34250922379628699801870512147, 7.05934061666323013527879908337, 8.148842058395079489925650181974, 9.083170531805857492149253801734, 10.33860557844077361248034349868, 11.91224628322178556279788578496, 13.34445354841232781348744816437, 13.99891486724709705679595640046