| L(s) = 1 | + (0.423 − 2.96i)3-s − 0.567i·5-s − 4.38·7-s + (−8.64 − 2.51i)9-s − 9.42i·11-s − 19.6·13-s + (−1.68 − 0.239i)15-s + 17.2i·17-s + 4.35·19-s + (−1.85 + 13.0i)21-s + 11.0i·23-s + 24.6·25-s + (−11.1 + 24.6i)27-s + 42.1i·29-s + 52.2·31-s + ⋯ |
| L(s) = 1 | + (0.141 − 0.989i)3-s − 0.113i·5-s − 0.627·7-s + (−0.960 − 0.279i)9-s − 0.857i·11-s − 1.51·13-s + (−0.112 − 0.0159i)15-s + 1.01i·17-s + 0.229·19-s + (−0.0884 + 0.620i)21-s + 0.478i·23-s + 0.987·25-s + (−0.411 + 0.911i)27-s + 1.45i·29-s + 1.68·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.141 - 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.141 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4709586864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4709586864\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.423 + 2.96i)T \) |
| 19 | \( 1 - 4.35T \) |
| good | 5 | \( 1 + 0.567iT - 25T^{2} \) |
| 7 | \( 1 + 4.38T + 49T^{2} \) |
| 11 | \( 1 + 9.42iT - 121T^{2} \) |
| 13 | \( 1 + 19.6T + 169T^{2} \) |
| 17 | \( 1 - 17.2iT - 289T^{2} \) |
| 23 | \( 1 - 11.0iT - 529T^{2} \) |
| 29 | \( 1 - 42.1iT - 841T^{2} \) |
| 31 | \( 1 - 52.2T + 961T^{2} \) |
| 37 | \( 1 + 53.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 23.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 13.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 40.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 30.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 103. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 88.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 11.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 35.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 70.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 119.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 39.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 95.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 55.3T + 9.40e3T^{2} \) |
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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
−10.11844911286977961997646258562, −9.021594151327149618187855180239, −8.450935870317310539974234130552, −7.42932449459534675064674284790, −6.77869313631350217770105891552, −5.90078934550229940473464258779, −4.96950991796457256546450608640, −3.44090925306109703136003214271, −2.62325848406391635538357134863, −1.24101307521852805238076447730,
0.15347988829755981689836645345, 2.43238039900209057117706849228, 3.15993611875622286441188704331, 4.59372772778220178064028490262, 4.91288254485000260463653076922, 6.25762556968267146469035254259, 7.15683443185853947270387348423, 8.071926985521246732485379783889, 9.168875262077762865588767769902, 9.871780498065777324936092918458
