L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.499 + 0.866i)6-s − 7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)10-s + 2·11-s + 0.999·12-s + (1.5 − 2.59i)13-s + (0.5 + 0.866i)14-s + (0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−2 − 3.46i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s + (−0.204 + 0.353i)6-s − 0.377·7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 − 0.273i)10-s + 0.603·11-s + 0.288·12-s + (0.416 − 0.720i)13-s + (0.133 + 0.231i)14-s + (0.129 − 0.223i)15-s + (−0.125 − 0.216i)16-s + (−0.485 − 0.840i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.710781 - 0.784003i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.710781 - 0.784003i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-4 - 1.73i)T \) |
good | 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + (-1.5 + 2.59i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5 + 8.66i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.5 - 4.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1 - 1.73i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.49141368337666326423582416115, −9.795021897069123410852039953444, −8.840088507065860518693972175046, −7.895621802286099964824644110513, −6.88981659299806373152911804805, −6.10634893588634323077240572864, −4.83999106076969052721781158083, −3.43438431865943422985268001112, −2.40450431160513422748603935096, −0.809873558801618410664511583089,
1.37480648026459237086670625954, 3.42636963520616200921620401874, 4.57929764095546905524834171997, 5.52483200491050289155150484852, 6.47373468576235738619693083457, 7.22744542919858744365105500972, 8.655238444588747556259208083972, 9.078593819957897214792129023348, 9.916436371647121268699902052222, 10.82968777311078803666024264811