| L(s) = 1 | + (−15.5 + 1.51i)3-s + (33.0 − 57.2i)5-s + (57.0 + 98.8i)7-s + (238. − 47.0i)9-s + (−192. − 333. i)11-s + (−516. + 894. i)13-s + (−425. + 938. i)15-s + 959.·17-s + 464.·19-s + (−1.03e3 − 1.44e3i)21-s + (1.15e3 − 1.99e3i)23-s + (−621. − 1.07e3i)25-s + (−3.62e3 + 1.09e3i)27-s + (−3.54e3 − 6.14e3i)29-s + (3.88e3 − 6.72e3i)31-s + ⋯ |
| L(s) = 1 | + (−0.995 + 0.0973i)3-s + (0.591 − 1.02i)5-s + (0.440 + 0.762i)7-s + (0.981 − 0.193i)9-s + (−0.480 − 0.832i)11-s + (−0.847 + 1.46i)13-s + (−0.488 + 1.07i)15-s + 0.804·17-s + 0.295·19-s + (−0.512 − 0.716i)21-s + (0.453 − 0.786i)23-s + (−0.198 − 0.344i)25-s + (−0.957 + 0.288i)27-s + (−0.783 − 1.35i)29-s + (0.725 − 1.25i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.359648225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.359648225\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (15.5 - 1.51i)T \) |
| good | 5 | \( 1 + (-33.0 + 57.2i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-57.0 - 98.8i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (192. + 333. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (516. - 894. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 - 959.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 464.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-1.15e3 + 1.99e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (3.54e3 + 6.14e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-3.88e3 + 6.72e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 9.31e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-6.66e3 + 1.15e4i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (1.05e3 + 1.82e3i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-1.24e3 - 2.16e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + 1.00e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-2.72e3 + 4.71e3i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.70e4 + 2.96e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.67e4 + 4.64e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 970.T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-1.60e4 - 2.78e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (1.80e4 + 3.12e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 - 4.26e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-2.19e4 - 3.79e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−11.93966866596790626184462640948, −11.22946716940901315716039226256, −9.809020432857191346625140848932, −9.090885639114381429649297897705, −7.75745226828008080615611668978, −6.17713883728918228153337828494, −5.36254105918634780912187016740, −4.44086976708107680366096734457, −2.09171955087252273463545846002, −0.60696312904251768375701243783,
1.19961456607917992183525497186, 2.95493188183095229471093207267, 4.80832002196870896628117968441, 5.74784827443496340345184592676, 7.13321720351444985001134290009, 7.63220160511949739208975793446, 9.881969110242975371053805881684, 10.34721942612174257170392816494, 11.14788424331191646338499577038, 12.40199241881152846156330170976
