L(s) = 1 | + (1.22 − 0.707i)2-s + (0.866 − 1.5i)3-s + (0.999 − 1.73i)4-s + (1.36 − 4.80i)5-s − 2.44i·6-s + (1.07 + 6.91i)7-s − 2.82i·8-s + (−1.5 − 2.59i)9-s + (−1.72 − 6.85i)10-s + (8.06 − 13.9i)11-s + (−1.73 − 2.99i)12-s − 2.01·13-s + (6.21 + 7.70i)14-s + (−6.03 − 6.21i)15-s + (−2.00 − 3.46i)16-s + (−1.22 + 2.12i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.288 − 0.5i)3-s + (0.249 − 0.433i)4-s + (0.273 − 0.961i)5-s − 0.408i·6-s + (0.153 + 0.988i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.172 − 0.685i)10-s + (0.732 − 1.26i)11-s + (−0.144 − 0.249i)12-s − 0.154·13-s + (0.443 + 0.550i)14-s + (−0.402 − 0.414i)15-s + (−0.125 − 0.216i)16-s + (−0.0722 + 0.125i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00461 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.00461 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.73616 - 1.72816i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.73616 - 1.72816i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 + 0.707i)T \) |
| 3 | \( 1 + (-0.866 + 1.5i)T \) |
| 5 | \( 1 + (-1.36 + 4.80i)T \) |
| 7 | \( 1 + (-1.07 - 6.91i)T \) |
good | 11 | \( 1 + (-8.06 + 13.9i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 2.01T + 169T^{2} \) |
| 17 | \( 1 + (1.22 - 2.12i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (8.06 - 4.65i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (22.3 - 12.9i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 37.0T + 841T^{2} \) |
| 31 | \( 1 + (-36.9 - 21.3i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-24.4 + 14.1i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 18.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 11.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-32.0 - 55.5i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-3.55 - 2.05i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-27.5 - 15.8i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (80.8 - 46.7i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-92.7 - 53.5i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 98.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + (27.6 - 47.8i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (59.3 + 102. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 88.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + (97.7 - 56.4i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 81.0T + 9.40e3T^{2} \) |
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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
−12.09502075079200752755115874950, −11.35167167270689472091602946720, −9.862316144559332818882507436014, −8.807727727378936678690224634870, −8.164014818908997666056306157160, −6.33354998340074331538234875328, −5.66280569519715768333053486307, −4.29533189425625513340884630717, −2.74494483417218569397471514321, −1.27735795001820304967492938051,
2.35804863812552814127555857408, 3.87003059631717272006576078897, 4.67736300106171535768067132902, 6.36935345270337523936384255124, 7.08721573327633780158292072911, 8.166280531501203801996471011236, 9.734982576051585589319420871708, 10.33351807691457531681842151891, 11.41248437466150072236343871145, 12.44343846955151259966078904181