L(s) = 1 | − 11.9i·2-s + 41.5i·3-s − 77.9·4-s − 111. i·5-s + 495.·6-s − 49.3i·7-s + 166. i·8-s − 998.·9-s − 1.32e3·10-s − 2.54e3·11-s − 3.24e3i·12-s − 277.·13-s − 587.·14-s + 4.63e3·15-s − 3.00e3·16-s − 532.·17-s + ⋯ |
L(s) = 1 | − 1.48i·2-s + 1.53i·3-s − 1.21·4-s − 0.892i·5-s + 2.29·6-s − 0.143i·7-s + 0.325i·8-s − 1.36·9-s − 1.32·10-s − 1.91·11-s − 1.87i·12-s − 0.126·13-s − 0.214·14-s + 1.37·15-s − 0.733·16-s − 0.108·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.364i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0956478 + 0.506582i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0956478 + 0.506582i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + (-7.40e4 - 2.89e4i)T \) |
good | 2 | \( 1 + 11.9iT - 64T^{2} \) |
| 3 | \( 1 - 41.5iT - 729T^{2} \) |
| 5 | \( 1 + 111. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 49.3iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 2.54e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 277.T + 4.82e6T^{2} \) |
| 17 | \( 1 + 532.T + 2.41e7T^{2} \) |
| 19 | \( 1 + 6.00e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.85e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + 2.70e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.61e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 2.17e3iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 8.26e4T + 4.75e9T^{2} \) |
| 47 | \( 1 + 1.68e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + 8.04e4T + 2.21e10T^{2} \) |
| 59 | \( 1 - 1.51e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 3.48e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 2.61e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 4.95e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.04e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 2.89e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 4.35e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 1.28e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 2.91e5T + 8.32e11T^{2} \) |
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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
−13.67057317417679786316006921088, −12.65104043474008496114012400393, −11.34495102406490405356618339013, −10.32654796837854653214683044334, −9.692914755822400801118564028727, −8.370826086087888199699643164888, −5.14813611496041841008526023422, −4.20093461180804021845925147401, −2.62946233664440245287316400799, −0.22607126320321661333934267650,
2.42183820657512897667728424528, 5.51497287460087394544791300464, 6.59693539430850476168865389937, 7.57632013563548740075652038288, 8.251895574245341395712496365918, 10.50267200859393755538902265304, 12.18422749072527676330205749609, 13.40945074303952024939658716419, 14.19618849400912782059295111069, 15.30718969012111741532678425963