L(s) = 1 | + (−0.5 + 0.866i)2-s + (1.37 − 1.05i)3-s + (−0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + (0.224 + 1.71i)6-s − 1.74·7-s + 0.999·8-s + (0.780 − 2.89i)9-s + (−0.866 + 0.499i)10-s + 4.48i·11-s + (−1.59 − 0.663i)12-s + (3.14 − 1.81i)13-s + (0.871 − 1.51i)14-s + (1.71 − 0.224i)15-s + (−0.5 + 0.866i)16-s + (5.72 + 3.30i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.793 − 0.608i)3-s + (−0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + (0.0917 + 0.701i)6-s − 0.659·7-s + 0.353·8-s + (0.260 − 0.965i)9-s + (−0.273 + 0.158i)10-s + 1.35i·11-s + (−0.461 − 0.191i)12-s + (0.872 − 0.503i)13-s + (0.232 − 0.403i)14-s + (0.443 − 0.0580i)15-s + (−0.125 + 0.216i)16-s + (1.38 + 0.800i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.65544 + 0.199439i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.65544 + 0.199439i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-1.37 + 1.05i)T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 19 | \( 1 + (-3.74 + 2.22i)T \) |
good | 7 | \( 1 + 1.74T + 7T^{2} \) |
| 11 | \( 1 - 4.48iT - 11T^{2} \) |
| 13 | \( 1 + (-3.14 + 1.81i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-5.72 - 3.30i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-1.69 + 0.977i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.271 + 0.469i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8.66iT - 31T^{2} \) |
| 37 | \( 1 - 0.251iT - 37T^{2} \) |
| 41 | \( 1 + (-2.28 + 3.95i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.13 - 3.70i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.28 + 0.743i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.0649 + 0.112i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.22 - 5.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.04 - 1.80i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.40 - 4.27i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.78 - 8.29i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.51 - 6.08i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.04 + 2.91i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 10.2iT - 83T^{2} \) |
| 89 | \( 1 + (2.10 + 3.64i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.01 - 5.20i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.31488185909547281717572395669, −9.743729731694036827959181158584, −9.013007817541225912425533255748, −7.927494788049267900212156865865, −7.32959685983587309611043289362, −6.41683904863122395476478586035, −5.58210099963160970900360073628, −3.98123609600267292544259654155, −2.78431180828351611596532750642, −1.31985031991689615234838928127,
1.34693849435748862282454699823, 3.15240487022106584360707685936, 3.42122723079917781561320519616, 4.96849972381003843005042906771, 6.01945581800616964784246364351, 7.42959239654982312054599199351, 8.416621690929310844204018331393, 9.089457960052512458457910065001, 9.751418776081630127443611355193, 10.52125714647738991493289794604