L(s) = 1 | + (−1 + i)2-s + (−2.35 − 1.86i)3-s − 2i·4-s + (−1.91 + 4.61i)5-s + (4.21 − 0.488i)6-s + (6.26 − 3.12i)7-s + (2 + 2i)8-s + (2.05 + 8.76i)9-s + (−2.70 − 6.53i)10-s + 0.117i·11-s + (−3.72 + 4.70i)12-s + (−9.72 − 9.72i)13-s + (−3.13 + 9.39i)14-s + (13.1 − 7.29i)15-s − 4·16-s + (−13.8 − 13.8i)17-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.5i)2-s + (−0.783 − 0.620i)3-s − 0.5i·4-s + (−0.383 + 0.923i)5-s + (0.702 − 0.0814i)6-s + (0.894 − 0.446i)7-s + (0.250 + 0.250i)8-s + (0.228 + 0.973i)9-s + (−0.270 − 0.653i)10-s + 0.0106i·11-s + (−0.310 + 0.391i)12-s + (−0.747 − 0.747i)13-s + (−0.223 + 0.670i)14-s + (0.873 − 0.486i)15-s − 0.250·16-s + (−0.815 − 0.815i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0746974 - 0.192907i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0746974 - 0.192907i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - i)T \) |
| 3 | \( 1 + (2.35 + 1.86i)T \) |
| 5 | \( 1 + (1.91 - 4.61i)T \) |
| 7 | \( 1 + (-6.26 + 3.12i)T \) |
good | 11 | \( 1 - 0.117iT - 121T^{2} \) |
| 13 | \( 1 + (9.72 + 9.72i)T + 169iT^{2} \) |
| 17 | \( 1 + (13.8 + 13.8i)T + 289iT^{2} \) |
| 19 | \( 1 + 29.7T + 361T^{2} \) |
| 23 | \( 1 + (-2.25 - 2.25i)T + 529iT^{2} \) |
| 29 | \( 1 + 46.0T + 841T^{2} \) |
| 31 | \( 1 - 1.50iT - 961T^{2} \) |
| 37 | \( 1 + (-5.32 - 5.32i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 13.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-36.8 + 36.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (29.7 + 29.7i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (59.8 + 59.8i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 84.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 34.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (34.4 + 34.4i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 77.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-41.3 - 41.3i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 0.865iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (99.0 - 99.0i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 129. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (15.7 - 15.7i)T - 9.40e3iT^{2} \) |
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
−11.39767648787919647233677737270, −10.95193047210870313123011777415, −10.05312069459990700724798014421, −8.428053680153044178112313061911, −7.44999596233507714443575898702, −6.93459433394496721619305764793, −5.68935547842031465906082344268, −4.45868180333766214539810120355, −2.18574760456272054094297227575, −0.14468719130353849226339487676,
1.82852286714088124826720478337, 4.13887324122572993106645291043, 4.79939856611963056535882142135, 6.20108371416085314070798713945, 7.76278121478456927819418519694, 8.848064100788085798047780661613, 9.460909312485527811644730629081, 10.88140563689324750702252491283, 11.31026259820112416891729285050, 12.35095329793307739362624881490