L(s) = 1 | + 11.3i·2-s − 143.·3-s − 128.·4-s − 309.·5-s − 1.62e3i·6-s + 1.25e3i·7-s − 1.44e3i·8-s + 1.39e4·9-s − 3.50e3i·10-s + (2.85e3 − 1.43e4i)11-s + 1.83e4·12-s + 3.85e4i·13-s − 1.42e4·14-s + 4.44e4·15-s + 1.63e4·16-s − 3.19e4i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 1.76·3-s − 0.500·4-s − 0.495·5-s − 1.25i·6-s + 0.523i·7-s − 0.353i·8-s + 2.12·9-s − 0.350i·10-s + (0.195 − 0.980i)11-s + 0.884·12-s + 1.35i·13-s − 0.370·14-s + 0.877·15-s + 0.250·16-s − 0.382i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.582923 - 0.0574745i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.582923 - 0.0574745i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 11.3iT \) |
| 11 | \( 1 + (-2.85e3 + 1.43e4i)T \) |
good | 3 | \( 1 + 143.T + 6.56e3T^{2} \) |
| 5 | \( 1 + 309.T + 3.90e5T^{2} \) |
| 7 | \( 1 - 1.25e3iT - 5.76e6T^{2} \) |
| 13 | \( 1 - 3.85e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 3.19e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.96e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 2.93e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 3.68e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 4.33e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 2.92e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.66e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 5.40e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 3.98e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 4.85e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 7.61e5T + 1.46e14T^{2} \) |
| 61 | \( 1 + 1.65e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 3.56e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 6.40e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + 1.18e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 5.34e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 1.61e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 1.55e6T + 3.93e15T^{2} \) |
| 97 | \( 1 + 1.42e8T + 7.83e15T^{2} \) |
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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
−16.32001254743607205648551548314, −15.32388180546903993259545716616, −13.42932235297281762386912470318, −11.85408422479473431947437722372, −11.14499142389372598064129361276, −9.162670140851231802511683781435, −7.07436326359773406932803814775, −5.91633112556146990266727982266, −4.54905278157928811308099504724, −0.48545729405894399113624797199,
1.02698248066841932738401131699, 4.15376144810945625957150341167, 5.65937990792877305618529010626, 7.49086413956371758220498480379, 10.06265930275647607234196107310, 10.90276149331917367583792989581, 12.12940280105085423635631921338, 12.89216540564802160315529820479, 15.07830011305514415146683728931, 16.55257181566500178361867281401