| L(s) = 1 | + (2.31 − 2.76i)2-s + (−1.23 − 3.38i)3-s + (−1.56 − 8.86i)4-s + (0.694 − 3.93i)5-s + (−12.2 − 4.44i)6-s + (2.5 + 4.33i)7-s + (−15.6 − 9.01i)8-s + (−3.06 + 2.57i)9-s + (−9.27 − 11.0i)10-s + (5 − 8.66i)11-s + (−28.1 + 16.2i)12-s + (−1.23 + 3.38i)13-s + (17.7 + 3.13i)14-s + (−14.2 + 2.50i)15-s + (−27.2 + 9.91i)16-s + (11.4 + 9.64i)17-s + ⋯ |
| L(s) = 1 | + (1.15 − 1.38i)2-s + (−0.411 − 1.12i)3-s + (−0.390 − 2.21i)4-s + (0.138 − 0.787i)5-s + (−2.03 − 0.741i)6-s + (0.357 + 0.618i)7-s + (−1.95 − 1.12i)8-s + (−0.340 + 0.285i)9-s + (−0.927 − 1.10i)10-s + (0.454 − 0.787i)11-s + (−2.34 + 1.35i)12-s + (−0.0948 + 0.260i)13-s + (1.26 + 0.223i)14-s + (−0.946 + 0.166i)15-s + (−1.70 + 0.619i)16-s + (0.675 + 0.567i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.932610 + 2.62401i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.932610 + 2.62401i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (-2.31 + 2.76i)T + (-0.694 - 3.93i)T^{2} \) |
| 3 | \( 1 + (1.23 + 3.38i)T + (-6.89 + 5.78i)T^{2} \) |
| 5 | \( 1 + (-0.694 + 3.93i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (-2.5 - 4.33i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-5 + 8.66i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (1.23 - 3.38i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-11.4 - 9.64i)T + (50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (-6.07 - 34.4i)T + (-497. + 180. i)T^{2} \) |
| 29 | \( 1 + (11.5 + 13.8i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-31.2 + 18.0i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 21.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-12.3 - 33.8i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (3.47 - 19.6i)T + (-1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-7.66 + 6.42i)T + (383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + (74.5 - 13.1i)T + (2.63e3 - 960. i)T^{2} \) |
| 59 | \( 1 + (-11.5 + 13.8i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (6.94 + 39.3i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (25.4 + 30.3i)T + (-779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (106. + 18.7i)T + (4.73e3 + 1.72e3i)T^{2} \) |
| 73 | \( 1 + (98.6 - 35.9i)T + (4.08e3 - 3.42e3i)T^{2} \) |
| 79 | \( 1 + (12.3 + 33.8i)T + (-4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-20 - 34.6i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-6.06e3 - 5.09e3i)T^{2} \) |
| 97 | \( 1 + (-78.7 + 93.9i)T + (-1.63e3 - 9.26e3i)T^{2} \) |
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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
−11.42725442560010647711729681548, −9.988023706410280451950836476258, −9.046564473623196185963463015383, −7.82403600042206152846669197297, −6.22727987305017310034701944633, −5.63791437334300190903386354819, −4.58489165731637373931893869190, −3.25375832667879245794009804057, −1.77904141298684921079288133804, −1.02860821635808078153962362708,
3.08487046249865727387176978684, 4.27711233543943523663953958757, 4.79270827217181916046754181716, 5.85787155661127648281376220747, 6.93275084165839481021935538415, 7.50174941002333234199905729132, 8.840915478877475509687453483398, 10.13086588694528591183145622655, 10.75639098434248487710262220951, 11.96077162768360924597093779493
