| L(s) = 1 | + (−1.22 + 1.22i)3-s + (−1.55 − 1.55i)7-s − 2.99i·9-s − 7.79·11-s + (2.44 − 2.44i)13-s + (8.89 + 8.89i)17-s + 5.59i·19-s + 3.79·21-s + (−2.20 + 2.20i)23-s + (3.67 + 3.67i)27-s − 35.5i·29-s − 53.1·31-s + (9.55 − 9.55i)33-s + (25.1 + 25.1i)37-s + 5.99i·39-s + ⋯ |
| L(s) = 1 | + (−0.408 + 0.408i)3-s + (−0.221 − 0.221i)7-s − 0.333i·9-s − 0.708·11-s + (0.188 − 0.188i)13-s + (0.523 + 0.523i)17-s + 0.294i·19-s + 0.180·21-s + (−0.0957 + 0.0957i)23-s + (0.136 + 0.136i)27-s − 1.22i·29-s − 1.71·31-s + (0.289 − 0.289i)33-s + (0.679 + 0.679i)37-s + 0.153i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.400885278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.400885278\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (1.55 + 1.55i)T + 49iT^{2} \) |
| 11 | \( 1 + 7.79T + 121T^{2} \) |
| 13 | \( 1 + (-2.44 + 2.44i)T - 169iT^{2} \) |
| 17 | \( 1 + (-8.89 - 8.89i)T + 289iT^{2} \) |
| 19 | \( 1 - 5.59iT - 361T^{2} \) |
| 23 | \( 1 + (2.20 - 2.20i)T - 529iT^{2} \) |
| 29 | \( 1 + 35.5iT - 841T^{2} \) |
| 31 | \( 1 + 53.1T + 961T^{2} \) |
| 37 | \( 1 + (-25.1 - 25.1i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 56.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-41.7 + 41.7i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-44.4 - 44.4i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-36.0 + 36.0i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 10.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 48.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-29.8 - 29.8i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 50.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + (49.3 - 49.3i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 66iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-4.09 + 4.09i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 29.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-133. - 133. i)T + 9.40e3iT^{2} \) |
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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
−9.759894887525315924844907149010, −8.842769050202271011463018827840, −7.889953989504713253819830205859, −7.18750468718611128425487670128, −5.96220422357060009234558304266, −5.53922475403133579530070022045, −4.32669412394470304656762295937, −3.55833140231100691265084448091, −2.29644299863557770380018414974, −0.70482936395277045562395804632,
0.72008924969337457855181175102, 2.13784420043216426377822309159, 3.18801950096158660486568416883, 4.44159227505069449618913390317, 5.47325326331912976793103726110, 6.03336595576211551181073267974, 7.24273908360537988138595178818, 7.60333823851982910308976779601, 8.821228570779328340288050478803, 9.414015655558716906497121398564
