L(s) = 1 | − 5.65·2-s + 3.84i·3-s + 32.0·4-s − 204. i·5-s − 21.7i·6-s + (−41.7 − 340. i)7-s − 181.·8-s + 714.·9-s + 1.15e3i·10-s − 1.17e3·11-s + 122. i·12-s + 1.39e3i·13-s + (236. + 1.92e3i)14-s + 786.·15-s + 1.02e3·16-s − 6.06e3i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.142i·3-s + 0.500·4-s − 1.63i·5-s − 0.100i·6-s + (−0.121 − 0.992i)7-s − 0.353·8-s + 0.979·9-s + 1.15i·10-s − 0.884·11-s + 0.0711i·12-s + 0.637i·13-s + (0.0861 + 0.701i)14-s + 0.233·15-s + 0.250·16-s − 1.23i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.121 + 0.992i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.712827 - 0.630670i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.712827 - 0.630670i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 5.65T \) |
| 7 | \( 1 + (41.7 + 340. i)T \) |
good | 3 | \( 1 - 3.84iT - 729T^{2} \) |
| 5 | \( 1 + 204. iT - 1.56e4T^{2} \) |
| 11 | \( 1 + 1.17e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 1.39e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 6.06e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 3.13e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 8.65e3T + 1.48e8T^{2} \) |
| 29 | \( 1 - 3.25e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 4.22e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 3.75e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 9.35e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 9.20e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 2.49e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 109.T + 2.21e10T^{2} \) |
| 59 | \( 1 - 3.26e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 8.45e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 1.11e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 4.41e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 4.64e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 5.95e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 6.54e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 2.98e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.69e5iT - 8.32e11T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.84229163890315113018370434649, −16.46931786760479609090359133281, −15.96364077901677901272266056601, −13.57004555493516136898908971881, −12.30240115825826189790306003323, −10.34238995854355790171013150365, −9.023600513172322870039378606323, −7.39285511371144148459613847782, −4.69914401864769701379118414498, −0.961920847447946679456680070997,
2.65164762697882211504464742228, 6.33859245696523188220522060595, 7.85411874414407392759396515208, 9.954250041488454821248368793856, 11.04606202617613967104475756779, 12.85915816698137519740319576869, 14.95232999854192493566121403808, 15.64724378542896190444631665480, 17.76389103660354726147601050344, 18.56442240284074113158727856741