L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s − 2·5-s + (0.499 − 0.866i)6-s + (0.5 − 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (1 + 1.73i)10-s + (−3.08 − 5.35i)11-s − 0.999·12-s + (−1.5 + 3.27i)13-s − 0.999·14-s + (−1 − 1.73i)15-s + (−0.5 − 0.866i)16-s + (−2.5 + 4.33i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s − 0.894·5-s + (0.204 − 0.353i)6-s + (0.188 − 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.316 + 0.547i)10-s + (−0.931 − 1.61i)11-s − 0.288·12-s + (−0.416 + 0.909i)13-s − 0.267·14-s + (−0.258 − 0.447i)15-s + (−0.125 − 0.216i)16-s + (−0.606 + 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0713627 - 0.389140i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0713627 - 0.389140i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (1.5 - 3.27i)T \) |
good | 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 + (3.08 + 5.35i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.08 + 7.08i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.58 + 2.75i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.08 + 7.08i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 7.17T + 31T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.08 + 3.61i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.410 - 0.711i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 4.17T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + (-0.410 + 0.711i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.58 - 13.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.58 - 7.94i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 1.82T + 79T^{2} \) |
| 83 | \( 1 + 5.17T + 83T^{2} \) |
| 89 | \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.17 - 10.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.63512173247268900056372132772, −9.503141534336360059508005702329, −8.675988821449877719630208959943, −8.011129274506011085100366665662, −7.09215554927866160353667894650, −5.56568902393654669087316917983, −4.33411656556602806139700567678, −3.59032881975647066795230539223, −2.39011307957279222375653869731, −0.23682064380232302450242243318,
1.90439736054073195820049045610, 3.41126679667870522850116032338, 4.85979907054153562112379120118, 5.63110288819011479644914281874, 7.23548071869529114342173401628, 7.49775028936306690129423511437, 8.190490692306051293303587545362, 9.382423608101691392648985535336, 10.06695633078835265168109457626, 11.18983502189237155005028573280