| L(s) = 1 | − 2.82i·2-s − 5.19·3-s − 8.00·4-s − 44.0·5-s + 14.6i·6-s + 46.8·7-s + 22.6i·8-s + 27·9-s + 124. i·10-s + 137. i·11-s + 41.5·12-s − 10.0i·13-s − 132. i·14-s + 228.·15-s + 64.0·16-s + 42.3·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.500·4-s − 1.76·5-s + 0.408i·6-s + 0.955·7-s + 0.353i·8-s + 0.333·9-s + 1.24i·10-s + 1.13i·11-s + 0.288·12-s − 0.0593i·13-s − 0.675i·14-s + 1.01·15-s + 0.250·16-s + 0.146·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.480i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.876 + 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4829426218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4829426218\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 + 5.19T \) |
| 59 | \( 1 + (1.67e3 + 3.05e3i)T \) |
| good | 5 | \( 1 + 44.0T + 625T^{2} \) |
| 7 | \( 1 - 46.8T + 2.40e3T^{2} \) |
| 11 | \( 1 - 137. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 10.0iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 42.3T + 8.35e4T^{2} \) |
| 19 | \( 1 - 73.9T + 1.30e5T^{2} \) |
| 23 | \( 1 + 110. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 404.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 2.67iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.80e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.94e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 669. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 1.47e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 5.20e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 1.78e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 5.77e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 1.55e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 3.04e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 5.84e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 3.17e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 5.24e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 7.13e3iT - 8.85e7T^{2} \) |
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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
−10.78980018765539929394584371391, −9.773889798498585014862311637739, −8.402575838173063584033425073190, −7.78365893741732999701574099620, −6.79422924677661716709013515669, −4.94500185102281855580721757029, −4.49940002846618439128295758670, −3.32028136941654447804649782493, −1.59318756766233616718306131809, −0.20907972270846564315140836540,
0.952289958460709348723837786845, 3.40457152682614440679516139951, 4.41222832307427016100938292028, 5.29463373928887828446960556015, 6.51821006882189797714839339239, 7.64116568933643660106069682821, 8.089076318327022248577273967582, 9.020947297284071295453541166565, 10.61397755337303982775530256094, 11.34325249624688309744489862538
