L(s) = 1 | − 7-s − 2·11-s + (1.5 − 2.59i)13-s + (2 + 3.46i)17-s + (−4 + 1.73i)19-s + (−2 + 3.46i)23-s + (2.5 − 4.33i)25-s + 3·31-s − 5·37-s + (2 + 3.46i)41-s + (−4.5 − 7.79i)43-s + (−5 + 8.66i)47-s − 6·49-s + (−2 + 3.46i)53-s + (7 + 12.1i)59-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 0.603·11-s + (0.416 − 0.720i)13-s + (0.485 + 0.840i)17-s + (−0.917 + 0.397i)19-s + (−0.417 + 0.722i)23-s + (0.5 − 0.866i)25-s + 0.538·31-s − 0.821·37-s + (0.312 + 0.541i)41-s + (−0.686 − 1.18i)43-s + (−0.729 + 1.26i)47-s − 0.857·49-s + (−0.274 + 0.475i)53-s + (0.911 + 1.57i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6471937214\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6471937214\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (4 - 1.73i)T \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 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 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + (-2 - 3.46i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.5 + 7.79i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5 - 8.66i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-7 - 12.1i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7 + 12.1i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 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
−9.011708647492945420120669559810, −8.213496544847131166094314044002, −7.81917103414825971954623642986, −6.70053956205680381026279992250, −6.04492871575234126600456227681, −5.34283757442321885089513625322, −4.30514794206535342587555458358, −3.45691490806085514991615931488, −2.57520345403634591764875152523, −1.33785728387919940032344961685,
0.20797960963389775676988670256, 1.74217892660147439893482688745, 2.80731273971856373700670131271, 3.66021506325547834898682859232, 4.70758701232997568595353666594, 5.33513026917926569138637212403, 6.50234408314565223826133130662, 6.80695692885730810265991100203, 7.896486269005897439567859813887, 8.518480271794223725325165149378