| L(s) = 1 | + (−4.76 + 1.54i)2-s + (10.9 + 7.95i)3-s + (−31.4 + 22.8i)4-s + (−55.7 + 171. i)5-s + (−64.4 − 20.9i)6-s + (91.1 + 125. i)7-s + (302. − 417. i)8-s + (−168. − 519. i)9-s − 902. i·10-s + (825. + 1.04e3i)11-s − 526.·12-s + (211. − 68.7i)13-s + (−628. − 456. i)14-s + (−1.97e3 + 1.43e3i)15-s + (−28.2 + 87.0i)16-s + (3.25e3 + 1.05e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.595 + 0.193i)2-s + (0.405 + 0.294i)3-s + (−0.491 + 0.357i)4-s + (−0.445 + 1.37i)5-s + (−0.298 − 0.0970i)6-s + (0.265 + 0.365i)7-s + (0.591 − 0.814i)8-s + (−0.231 − 0.711i)9-s − 0.902i·10-s + (0.620 + 0.784i)11-s − 0.304·12-s + (0.0962 − 0.0312i)13-s + (−0.228 − 0.166i)14-s + (−0.585 + 0.425i)15-s + (−0.00690 + 0.0212i)16-s + (0.662 + 0.215i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.497951 + 0.732804i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.497951 + 0.732804i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-825. - 1.04e3i)T \) |
| good | 2 | \( 1 + (4.76 - 1.54i)T + (51.7 - 37.6i)T^{2} \) |
| 3 | \( 1 + (-10.9 - 7.95i)T + (225. + 693. i)T^{2} \) |
| 5 | \( 1 + (55.7 - 171. i)T + (-1.26e4 - 9.18e3i)T^{2} \) |
| 7 | \( 1 + (-91.1 - 125. i)T + (-3.63e4 + 1.11e5i)T^{2} \) |
| 13 | \( 1 + (-211. + 68.7i)T + (3.90e6 - 2.83e6i)T^{2} \) |
| 17 | \( 1 + (-3.25e3 - 1.05e3i)T + (1.95e7 + 1.41e7i)T^{2} \) |
| 19 | \( 1 + (4.58e3 - 6.30e3i)T + (-1.45e7 - 4.47e7i)T^{2} \) |
| 23 | \( 1 - 1.49e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-1.60e4 - 2.21e4i)T + (-1.83e8 + 5.65e8i)T^{2} \) |
| 31 | \( 1 + (6.31e3 + 1.94e4i)T + (-7.18e8 + 5.21e8i)T^{2} \) |
| 37 | \( 1 + (6.85e4 - 4.98e4i)T + (7.92e8 - 2.44e9i)T^{2} \) |
| 41 | \( 1 + (-6.34e4 + 8.73e4i)T + (-1.46e9 - 4.51e9i)T^{2} \) |
| 43 | \( 1 + 9.23e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (6.97e4 + 5.06e4i)T + (3.33e9 + 1.02e10i)T^{2} \) |
| 53 | \( 1 + (-3.61e4 - 1.11e5i)T + (-1.79e10 + 1.30e10i)T^{2} \) |
| 59 | \( 1 + (7.10e4 - 5.16e4i)T + (1.30e10 - 4.01e10i)T^{2} \) |
| 61 | \( 1 + (-3.35e5 - 1.09e5i)T + (4.16e10 + 3.02e10i)T^{2} \) |
| 67 | \( 1 + 2.36e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (-4.10e4 + 1.26e5i)T + (-1.03e11 - 7.52e10i)T^{2} \) |
| 73 | \( 1 + (-2.02e5 - 2.78e5i)T + (-4.67e10 + 1.43e11i)T^{2} \) |
| 79 | \( 1 + (-5.48e5 + 1.78e5i)T + (1.96e11 - 1.42e11i)T^{2} \) |
| 83 | \( 1 + (5.63e5 + 1.83e5i)T + (2.64e11 + 1.92e11i)T^{2} \) |
| 89 | \( 1 - 2.71e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (1.54e5 + 4.75e5i)T + (-6.73e11 + 4.89e11i)T^{2} \) |
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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
−19.26083210772306999311708049965, −18.33777968225104755889167059055, −17.15747471462209821784136453634, −15.23739752934387071599898433183, −14.37235957242609814573626078617, −12.17877411325755356220586806135, −10.25851236149907678789044801523, −8.722486577211605727580591125199, −7.03967980757278552225807536113, −3.61467095274975142542907188803,
0.970168928955669450792746637310, 4.86110123504744287004218098366, 8.127320725736841030611380117629, 9.071407689820625758638835062014, 11.11167697442146970036314219919, 13.06313893618964471594883640167, 14.25406315477464182782473850160, 16.38269986435277310205744541439, 17.39806317334033830885122203947, 19.25218486048643853674030251949
