L(s) = 1 | + (−1.07 − 0.621i)2-s + (26.6 + 4.11i)3-s + (−31.2 − 54.0i)4-s + (93.7 + 54.1i)5-s + (−26.1 − 21.0i)6-s + (147. − 309. i)7-s + 157. i·8-s + (695. + 219. i)9-s + (−67.2 − 116. i)10-s + (1.89e3 − 1.09e3i)11-s + (−610. − 1.57e3i)12-s − 1.13e3·13-s + (−351. + 241. i)14-s + (2.27e3 + 1.83e3i)15-s + (−1.90e3 + 3.29e3i)16-s + (−6.94e3 + 4.01e3i)17-s + ⋯ |
L(s) = 1 | + (−0.134 − 0.0776i)2-s + (0.988 + 0.152i)3-s + (−0.487 − 0.845i)4-s + (0.749 + 0.433i)5-s + (−0.121 − 0.0972i)6-s + (0.431 − 0.902i)7-s + 0.306i·8-s + (0.953 + 0.301i)9-s + (−0.0672 − 0.116i)10-s + (1.42 − 0.822i)11-s + (−0.353 − 0.909i)12-s − 0.518·13-s + (−0.128 + 0.0878i)14-s + (0.675 + 0.542i)15-s + (−0.464 + 0.803i)16-s + (−1.41 + 0.816i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.510i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.859 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.91597 - 0.525781i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.91597 - 0.525781i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-26.6 - 4.11i)T \) |
| 7 | \( 1 + (-147. + 309. i)T \) |
good | 2 | \( 1 + (1.07 + 0.621i)T + (32 + 55.4i)T^{2} \) |
| 5 | \( 1 + (-93.7 - 54.1i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-1.89e3 + 1.09e3i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + 1.13e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + (6.94e3 - 4.01e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (3.23e3 - 5.60e3i)T + (-2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-1.11e3 - 642. i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 - 1.42e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + (8.12e3 + 1.40e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (1.66e4 - 2.88e4i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 - 9.86e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 6.21e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (5.03e4 + 2.90e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-1.57e5 + 9.07e4i)T + (1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (1.27e5 - 7.36e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-6.34e3 + 1.09e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.88e5 + 3.26e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 1.18e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + (-3.41e3 - 5.91e3i)T + (-7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (1.83e5 - 3.18e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 - 3.36e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (2.58e5 + 1.49e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + 5.39e5T + 8.32e11T^{2} \) |
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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
−16.90436967899541024421370995086, −14.89022266419430283401667087914, −14.23124812665722311021018100624, −13.38960286810668241890333401400, −10.87004799822555233065875296731, −9.793226120587080500461436294553, −8.569088263733361217151263502403, −6.47766357304740580504230923033, −4.18096523334168744654207269091, −1.64572174899549192799018323268,
2.22462525236433946725546285920, 4.48030527365482307768738654743, 7.10429105552012503605099913537, 8.942885450806257581088983923313, 9.253960138032233117195606107673, 12.01354386525823759371149796483, 13.10575211663651481315130142464, 14.24875611886373423721609206265, 15.53621175650252969721151866625, 17.30641158794401892795989632530