| L(s) = 1 | + 22.6·2-s + (137. + 200. i)3-s + 512.·4-s + (3.11e3 + 4.53e3i)6-s − 2.32e4i·7-s + 1.15e4·8-s + (−2.11e4 + 5.51e4i)9-s − 6.24e4i·11-s + (7.04e4 + 1.02e5i)12-s + 1.70e5i·13-s − 5.25e5i·14-s + 2.62e5·16-s − 2.66e6·17-s + (−4.78e5 + 1.24e6i)18-s − 7.66e5·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + (0.566 + 0.824i)3-s + 0.500·4-s + (0.400 + 0.582i)6-s − 1.38i·7-s + 0.353·8-s + (−0.358 + 0.933i)9-s − 0.387i·11-s + (0.283 + 0.412i)12-s + 0.458i·13-s − 0.977i·14-s + 0.250·16-s − 1.87·17-s + (−0.253 + 0.660i)18-s − 0.309·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.5513398234\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5513398234\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 22.6T \) |
| 3 | \( 1 + (-137. - 200. i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2.32e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 6.24e4iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 1.70e5iT - 1.37e11T^{2} \) |
| 17 | \( 1 + 2.66e6T + 2.01e12T^{2} \) |
| 19 | \( 1 + 7.66e5T + 6.13e12T^{2} \) |
| 23 | \( 1 + 1.40e6T + 4.14e13T^{2} \) |
| 29 | \( 1 - 4.83e6iT - 4.20e14T^{2} \) |
| 31 | \( 1 + 4.18e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 5.01e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 - 1.49e8iT - 1.34e16T^{2} \) |
| 43 | \( 1 - 1.98e8iT - 2.16e16T^{2} \) |
| 47 | \( 1 + 1.55e8T + 5.25e16T^{2} \) |
| 53 | \( 1 + 4.21e7T + 1.74e17T^{2} \) |
| 59 | \( 1 - 2.92e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 5.30e8T + 7.13e17T^{2} \) |
| 67 | \( 1 - 5.22e8iT - 1.82e18T^{2} \) |
| 71 | \( 1 + 5.71e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 2.18e9iT - 4.29e18T^{2} \) |
| 79 | \( 1 + 1.96e9T + 9.46e18T^{2} \) |
| 83 | \( 1 - 2.18e9T + 1.55e19T^{2} \) |
| 89 | \( 1 + 2.38e8iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 8.84e9iT - 7.37e19T^{2} \) |
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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
−11.28931259860079783358408167104, −10.83842767736000125182628154859, −9.736694505519952359873095035237, −8.596404267419198211145566177266, −7.41061745181519636650412299348, −6.33181742670421847865387515149, −4.72292170707866090903691634537, −4.14551342916370424380454137687, −3.08634921290407597368939515914, −1.72337831717102526372073017489,
0.07249611009353107557283541923, 1.95742752735735317643691040819, 2.44347211532622590679067670518, 3.81540857071461933515989233314, 5.30698782514529232304721321753, 6.29538234680419666415187175863, 7.27637772857508696863309042454, 8.535484478798877967346521359848, 9.241454489003880345445239512123, 10.90129628874669612030330792992
