| L(s) = 1 | + (−0.996 − 0.0871i)2-s + (0.984 + 0.173i)4-s + (−2.02 + 0.946i)5-s + (−2.76 − 1.93i)7-s + (−0.965 − 0.258i)8-s + (2.10 − 0.766i)10-s + (−0.0536 + 0.147i)11-s + (0.614 + 7.02i)13-s + (2.58 + 2.16i)14-s + (0.939 + 0.342i)16-s + (2.66 − 0.713i)17-s + (4.34 − 2.51i)19-s + (−2.15 + 0.580i)20-s + (0.0662 − 0.142i)22-s + (−4.32 − 6.16i)23-s + ⋯ |
| L(s) = 1 | + (−0.704 − 0.0616i)2-s + (0.492 + 0.0868i)4-s + (−0.906 + 0.423i)5-s + (−1.04 − 0.731i)7-s + (−0.341 − 0.0915i)8-s + (0.664 − 0.242i)10-s + (−0.0161 + 0.0444i)11-s + (0.170 + 1.94i)13-s + (0.691 + 0.579i)14-s + (0.234 + 0.0855i)16-s + (0.645 − 0.173i)17-s + (0.997 − 0.575i)19-s + (−0.482 + 0.129i)20-s + (0.0141 − 0.0303i)22-s + (−0.900 − 1.28i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.624444 - 0.288338i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.624444 - 0.288338i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.996 + 0.0871i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.02 - 0.946i)T \) |
| good | 7 | \( 1 + (2.76 + 1.93i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (0.0536 - 0.147i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.614 - 7.02i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-2.66 + 0.713i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-4.34 + 2.51i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.32 + 6.16i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-0.210 + 0.176i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.75 + 9.96i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.581 - 2.17i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-4.27 + 5.09i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.49 - 3.20i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-1.25 + 1.79i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-4.14 + 4.14i)T - 53iT^{2} \) |
| 59 | \( 1 + (-8.92 + 3.24i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.06 - 6.05i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (5.61 - 0.491i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-6.34 - 3.66i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.43 + 12.8i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (4.50 + 5.36i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.118 - 1.35i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-2.96 - 5.13i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.50 - 1.16i)T + (62.3 - 74.3i)T^{2} \) |
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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
−9.974183803859158337592078972131, −9.453873638582612706636148853679, −8.447370559623600539996263422934, −7.45569931849288557105109409593, −6.90084619820301674556517650028, −6.15610616594711372177973891399, −4.38282956134282374006223040196, −3.66149181185339014720974045258, −2.44021501757072019697402045258, −0.55731478826393179252582056208,
1.00517783509614299608072607280, 2.97806104313505984317614094870, 3.58565864613831114339050305063, 5.36808120139981090919516123688, 5.87876579766466697234563541001, 7.22042473460065274304996118103, 7.936328512733891490555126258393, 8.560947773614856318232355689280, 9.574134728101524225399451513147, 10.15014787768588646258333957772
