L(s) = 1 | + (1.15 + 1.63i)2-s + 1.73i·3-s + (−1.32 + 3.77i)4-s − 5.34·5-s + (−2.82 + 2.00i)6-s − 12.9i·7-s + (−7.68 + 2.21i)8-s − 2.99·9-s + (−6.18 − 8.71i)10-s + 7.61i·11-s + (−6.53 − 2.28i)12-s − 20.9·13-s + (21.2 − 15.0i)14-s − 9.25i·15-s + (−12.5 − 9.97i)16-s − 4.33·17-s + ⋯ |
L(s) = 1 | + (0.578 + 0.815i)2-s + 0.577i·3-s + (−0.330 + 0.943i)4-s − 1.06·5-s + (−0.470 + 0.334i)6-s − 1.85i·7-s + (−0.960 + 0.276i)8-s − 0.333·9-s + (−0.618 − 0.871i)10-s + 0.691i·11-s + (−0.544 − 0.190i)12-s − 1.60·13-s + (1.51 − 1.07i)14-s − 0.616i·15-s + (−0.781 − 0.623i)16-s − 0.254·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 + 0.943i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0706428 - 0.0995628i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0706428 - 0.0995628i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.15 - 1.63i)T \) |
| 3 | \( 1 - 1.73iT \) |
| 23 | \( 1 - 4.79iT \) |
good | 5 | \( 1 + 5.34T + 25T^{2} \) |
| 7 | \( 1 + 12.9iT - 49T^{2} \) |
| 11 | \( 1 - 7.61iT - 121T^{2} \) |
| 13 | \( 1 + 20.9T + 169T^{2} \) |
| 17 | \( 1 + 4.33T + 289T^{2} \) |
| 19 | \( 1 - 4.70iT - 361T^{2} \) |
| 29 | \( 1 - 33.9T + 841T^{2} \) |
| 31 | \( 1 + 11.3iT - 961T^{2} \) |
| 37 | \( 1 - 8.17T + 1.36e3T^{2} \) |
| 41 | \( 1 + 64.9T + 1.68e3T^{2} \) |
| 43 | \( 1 - 65.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 62.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 1.27T + 2.80e3T^{2} \) |
| 59 | \( 1 + 38.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 80.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 12.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 65.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 89.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 127. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 125. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 34.7T + 7.92e3T^{2} \) |
| 97 | \( 1 - 142.T + 9.40e3T^{2} \) |
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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
−12.34687463026389610468028043217, −11.54445362222804372982817117147, −10.38186111382207405526638219223, −9.517580973107445644098679208553, −7.941316062941316218263557227368, −7.51616843711508916581090208423, −6.60166288176956595153486442623, −4.70222074679734254402932280680, −4.41147456468141795905655429480, −3.24867035115860252721004772077,
0.04799665739992655757169509022, 2.23005333065100768292575298226, 3.15833715419397369831896196441, 4.77595343830089140242089215303, 5.67710242627587625794326587288, 6.89823705055945209959630841609, 8.334545248272329454584054885481, 8.996142497142212699659621179706, 10.25493512154546639485974166282, 11.57994649878457517100714340002