L(s) = 1 | + 1.17·2-s + 9.10i·3-s − 14.6·4-s + 11.1i·5-s + 10.6i·6-s + (−40.5 + 27.5i)7-s − 35.9·8-s − 1.87·9-s + 13.1i·10-s + 106.·11-s − 133. i·12-s + 243. i·13-s + (−47.5 + 32.3i)14-s − 101.·15-s + 191.·16-s − 148. i·17-s + ⋯ |
L(s) = 1 | + 0.293·2-s + 1.01i·3-s − 0.913·4-s + 0.447i·5-s + 0.296i·6-s + (−0.826 + 0.562i)7-s − 0.561·8-s − 0.0231·9-s + 0.131i·10-s + 0.882·11-s − 0.924i·12-s + 1.43i·13-s + (−0.242 + 0.165i)14-s − 0.452·15-s + 0.749·16-s − 0.512i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.527671 + 0.997766i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.527671 + 0.997766i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - 11.1iT \) |
| 7 | \( 1 + (40.5 - 27.5i)T \) |
good | 2 | \( 1 - 1.17T + 16T^{2} \) |
| 3 | \( 1 - 9.10iT - 81T^{2} \) |
| 11 | \( 1 - 106.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 243. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 148. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 244. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 621.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 774.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 403. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.29e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.53e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.24e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.49e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 281.T + 7.89e6T^{2} \) |
| 59 | \( 1 + 5.82e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 6.26e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 4.12e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 793.T + 2.54e7T^{2} \) |
| 73 | \( 1 + 1.60e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 9.85e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 617. iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 7.34e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 67.6iT - 8.85e7T^{2} \) |
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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
−16.04874658594710082500866030796, −14.95817595953568866956050631528, −14.00317722463341272851007355457, −12.66926849913048319293804949473, −11.22534652834666613787050653274, −9.472653649293042801833040397919, −9.209524762779134939462511394657, −6.64328634492241439247201969979, −4.83013286727816391897178659131, −3.51599073321578748746783719278,
0.798391254824136071476031379032, 3.79427199849119524054647088234, 5.78131771745191043063098567108, 7.37171025043010950617173424372, 8.835551765556975416027430990784, 10.21992354529943637597600609945, 12.35181653281157088395893898906, 12.94251272915953340970913531935, 13.77198477773281493640313458290, 15.17151033356563211842518504787