L(s) = 1 | + (−1.91 − 0.589i)2-s − 1.73i·3-s + (3.30 + 2.25i)4-s + 5.58i·5-s + (−1.02 + 3.31i)6-s − 6.49i·7-s + (−4.98 − 6.25i)8-s − 2.99·9-s + (3.29 − 10.6i)10-s + (9.14 + 6.11i)11-s + (3.90 − 5.72i)12-s + 0.123·13-s + (−3.82 + 12.4i)14-s + 9.67·15-s + (5.84 + 14.8i)16-s + 22.9i·17-s + ⋯ |
L(s) = 1 | + (−0.955 − 0.294i)2-s − 0.577i·3-s + (0.826 + 0.563i)4-s + 1.11i·5-s + (−0.170 + 0.551i)6-s − 0.927i·7-s + (−0.623 − 0.781i)8-s − 0.333·9-s + (0.329 − 1.06i)10-s + (0.831 + 0.555i)11-s + (0.325 − 0.477i)12-s + 0.00946·13-s + (−0.273 + 0.886i)14-s + 0.645·15-s + (0.365 + 0.930i)16-s + 1.34i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0836i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.05140 + 0.0440282i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05140 + 0.0440282i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.91 + 0.589i)T \) |
| 3 | \( 1 + 1.73iT \) |
| 11 | \( 1 + (-9.14 - 6.11i)T \) |
good | 5 | \( 1 - 5.58iT - 25T^{2} \) |
| 7 | \( 1 + 6.49iT - 49T^{2} \) |
| 13 | \( 1 - 0.123T + 169T^{2} \) |
| 17 | \( 1 - 22.9iT - 289T^{2} \) |
| 19 | \( 1 + 9.48T + 361T^{2} \) |
| 23 | \( 1 - 29.5T + 529T^{2} \) |
| 29 | \( 1 - 22.1T + 841T^{2} \) |
| 31 | \( 1 - 42.8T + 961T^{2} \) |
| 37 | \( 1 + 18.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 50.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 11.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 20.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 65.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 77.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 35.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 12.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 102.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 4.28iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 37.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 95.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 104.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 168.T + 9.40e3T^{2} \) |
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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
−11.50436040438125618101307169228, −10.66523495849806414619482628145, −10.10145720338037411635174073349, −8.820657681811151319599841016318, −7.78396011289710017023368326979, −6.84185363930189509407094896288, −6.42211301064899624493477097782, −4.02792719205621473752307811058, −2.73810374344139345588915773758, −1.24246762025785547706176571911,
0.896446699092248718745187045135, 2.76642246744892682543609838474, 4.72266337341652532416515049399, 5.64381663853097894709611489544, 6.79348826091329693951975092141, 8.283292544825970884626530927418, 8.980547176399269772032563923466, 9.374053847990362909503234500110, 10.62932737810492801203158238392, 11.69653497978727050171540004923