L(s) = 1 | + (0.5 − 0.866i)2-s − 0.571·3-s + (−0.499 − 0.866i)4-s + 5-s + (−0.285 + 0.495i)6-s + (−2.12 − 3.67i)7-s − 0.999·8-s − 2.67·9-s + (0.5 − 0.866i)10-s + (−0.336 − 0.582i)11-s + (0.285 + 0.495i)12-s + (−2.12 + 3.67i)13-s − 4.24·14-s − 0.571·15-s + (−0.5 + 0.866i)16-s + (−1.05 + 1.81i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s − 0.330·3-s + (−0.249 − 0.433i)4-s + 0.447·5-s + (−0.116 + 0.202i)6-s + (−0.802 − 1.38i)7-s − 0.353·8-s − 0.890·9-s + (0.158 − 0.273i)10-s + (−0.101 − 0.175i)11-s + (0.0825 + 0.142i)12-s + (−0.588 + 1.01i)13-s − 1.13·14-s − 0.147·15-s + (−0.125 + 0.216i)16-s + (−0.254 + 0.441i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.344i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0995006 + 0.560368i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0995006 + 0.560368i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (-2.00 + 7.93i)T \) |
good | 3 | \( 1 + 0.571T + 3T^{2} \) |
| 7 | \( 1 + (2.12 + 3.67i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.336 + 0.582i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.12 - 3.67i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.05 - 1.81i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.12 - 1.94i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.45 + 7.72i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.42 + 4.20i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.52 - 7.84i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.33 + 5.77i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 0.571T + 43T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 0.899T + 53T^{2} \) |
| 59 | \( 1 - 3.96T + 59T^{2} \) |
| 61 | \( 1 + (-5.67 + 9.82i)T + (-30.5 - 52.8i)T^{2} \) |
| 71 | \( 1 + (-3.78 - 6.55i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.05 - 3.55i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.81 + 3.14i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.480 - 0.832i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 7.20T + 89T^{2} \) |
| 97 | \( 1 + (1.52 - 2.63i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.00237920049278224849297532025, −9.598336277940621128642251531201, −8.428077763376790428113543857542, −7.13722863349453041169059899013, −6.37743056663504163893490305154, −5.43014077171549889497946358913, −4.27530377402104347160313070222, −3.39316063216679364977496432433, −2.03932702877338548570707409551, −0.25874047466835349840195160082,
2.47884610247343454185732865652, 3.30882509336451798676502847616, 5.14594883614172097511375128260, 5.51766782395797704190411070164, 6.33467441363124038359729140721, 7.30019356903191626916415185263, 8.526070560240589380794321951306, 9.075226023741075388009654896437, 9.997994991681606832227847380693, 11.04006492109665470134250788921