L(s) = 1 | + (−2 − 2i)2-s + (6.67 − 6.67i)3-s + 8i·4-s + (5.03 − 24.4i)5-s − 26.6·6-s + (−43.7 − 43.7i)7-s + (16 − 16i)8-s − 8.02i·9-s + (−59.0 + 38.8i)10-s − 158.·11-s + (53.3 + 53.3i)12-s + (−13.0 + 13.0i)13-s + 175. i·14-s + (−129. − 196. i)15-s − 64·16-s + (108. + 108. i)17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.5i)2-s + (0.741 − 0.741i)3-s + 0.5i·4-s + (0.201 − 0.979i)5-s − 0.741·6-s + (−0.892 − 0.892i)7-s + (0.250 − 0.250i)8-s − 0.0990i·9-s + (−0.590 + 0.388i)10-s − 1.30·11-s + (0.370 + 0.370i)12-s + (−0.0773 + 0.0773i)13-s + 0.892i·14-s + (−0.576 − 0.875i)15-s − 0.250·16-s + (0.375 + 0.375i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4793619908\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4793619908\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2 + 2i)T \) |
| 5 | \( 1 + (-5.03 + 24.4i)T \) |
| 23 | \( 1 + (77.9 - 77.9i)T \) |
good | 3 | \( 1 + (-6.67 + 6.67i)T - 81iT^{2} \) |
| 7 | \( 1 + (43.7 + 43.7i)T + 2.40e3iT^{2} \) |
| 11 | \( 1 + 158.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (13.0 - 13.0i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (-108. - 108. i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 - 274. iT - 1.30e5T^{2} \) |
| 29 | \( 1 + 869. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 77.4T + 9.23e5T^{2} \) |
| 37 | \( 1 + (1.18e3 + 1.18e3i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 + 428.T + 2.82e6T^{2} \) |
| 43 | \( 1 + (-601. + 601. i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (-1.75e3 - 1.75e3i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (52.8 - 52.8i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 - 3.67e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 4.01e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-4.08e3 - 4.08e3i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 + 9.04e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (4.87e3 - 4.87e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 + 1.00e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (2.99e3 - 2.99e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 - 784. iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (1.07e4 + 1.07e4i)T + 8.85e7iT^{2} \) |
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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
−10.55785212967920150668472757719, −9.950475234434193487493096860145, −8.829320623459969338256953312017, −7.938853898234510586098321704873, −7.30058471521235123796534728091, −5.71169264812732232473421167166, −4.12171293220227689807312070329, −2.76389472444940420415396806128, −1.51079428331813919271835042115, −0.16566710622053209292029436759,
2.55272159199291704516394824483, 3.26616333221163108952381286678, 5.11026164452639600363849692192, 6.24400810069874039293883021241, 7.26740316447811145296983318446, 8.458027657391581937785567828084, 9.343440983840233033774349262310, 10.04668702765608399810889701361, 10.78796610115076240321294134093, 12.18551325637775240517156721143