L(s) = 1 | + (1.5 − 2.59i)5-s + (1 + 1.73i)7-s + (3 + 5.19i)11-s + (−2.5 + 4.33i)13-s + 3·17-s − 2·19-s + (−3 + 5.19i)23-s + (−2 − 3.46i)25-s + (1.5 + 2.59i)29-s + (−2 + 3.46i)31-s + 6·35-s + 5·37-s + (−3 + 5.19i)41-s + (−5 − 8.66i)43-s + (1.50 − 2.59i)49-s + ⋯ |
L(s) = 1 | + (0.670 − 1.16i)5-s + (0.377 + 0.654i)7-s + (0.904 + 1.56i)11-s + (−0.693 + 1.20i)13-s + 0.727·17-s − 0.458·19-s + (−0.625 + 1.08i)23-s + (−0.400 − 0.692i)25-s + (0.278 + 0.482i)29-s + (−0.359 + 0.622i)31-s + 1.01·35-s + 0.821·37-s + (−0.468 + 0.811i)41-s + (−0.762 − 1.32i)43-s + (0.214 − 0.371i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.910078438\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.910078438\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5 + 8.66i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.513145508856414130670156542305, −9.162759996334966976678014034875, −8.291946615695808680903976279477, −7.24754062810363543825656749735, −6.45656483339695475062935820303, −5.29783298288663241460452518615, −4.83794961968080808395419789308, −3.87342366276170379230902321836, −2.06435114104557519694871934833, −1.57573463571160337084355565395,
0.838348582054009283819936269622, 2.45428163444717510897293725720, 3.28414556135854286180528628589, 4.28288968474340840483283913036, 5.68008720901731402825000360580, 6.15968183716336270652544796730, 7.05968156440363707028324450756, 7.928543513752632560381582392022, 8.653957286510925976905740786362, 9.877865463999594368227943876935