| L(s) = 1 | + (0.766 − 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (−0.939 + 0.342i)6-s + (0.733 − 1.27i)7-s + (−0.500 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.766 − 0.642i)10-s + (−1.67 − 2.89i)11-s + (−0.499 + 0.866i)12-s + (2.09 − 0.761i)13-s + (−0.254 − 1.44i)14-s + (−0.173 + 0.984i)15-s + (−0.939 − 0.342i)16-s + (−2.43 + 2.04i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−0.542 − 0.197i)3-s + (0.0868 − 0.492i)4-s + (−0.0776 − 0.440i)5-s + (−0.383 + 0.139i)6-s + (0.277 − 0.480i)7-s + (−0.176 − 0.306i)8-s + (0.255 + 0.214i)9-s + (−0.242 − 0.203i)10-s + (−0.504 − 0.874i)11-s + (−0.144 + 0.250i)12-s + (0.580 − 0.211i)13-s + (−0.0681 − 0.386i)14-s + (−0.0448 + 0.254i)15-s + (−0.234 − 0.0855i)16-s + (−0.591 + 0.496i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.464489 - 1.27747i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.464489 - 1.27747i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (0.939 + 0.342i)T \) |
| 5 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (4.07 - 1.55i)T \) |
| good | 7 | \( 1 + (-0.733 + 1.27i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.67 + 2.89i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.09 + 0.761i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (2.43 - 2.04i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.745 + 4.22i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (5.08 + 4.26i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-5.40 + 9.35i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.02T + 37T^{2} \) |
| 41 | \( 1 + (-2.93 - 1.06i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.578 - 3.28i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (0.592 + 0.497i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (1.30 - 7.42i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-3.37 + 2.83i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.16 + 6.60i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-10.6 - 8.97i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (1.47 + 8.36i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-13.0 - 4.75i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.49 - 0.543i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (0.397 - 0.689i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-10.2 + 3.72i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-5.36 + 4.49i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.87192714484081869625606658536, −9.769494835255100013957208087185, −8.533446423404496183541876282883, −7.81074588191774314758645524264, −6.38978788869010806294110252193, −5.81224722977403824751438068889, −4.62238544235471097394706443349, −3.84304940056845913339519240997, −2.24979102130868326308102986062, −0.68739039535476345136347973907,
2.15078941611350843788293660605, 3.57277847224967723669613116260, 4.75309695404728942923969092000, 5.43608379323517801041077236541, 6.61169829014863446594716833438, 7.16585068024135144531635572826, 8.364524311906813254112974388120, 9.255494447862818094078338902907, 10.42554288456245815430491055721, 11.13941952674962185883281190997
