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
−11.18983502189237155005028573280, −10.06695633078835265168109457626, −9.382423608101691392648985535336, −8.190490692306051293303587545362, −7.49775028936306690129423511437, −7.23548071869529114342173401628, −5.63110288819011479644914281874, −4.85979907054153562112379120118, −3.41126679667870522850116032338, −1.90439736054073195820049045610,
0.23682064380232302450242243318, 2.39011307957279222375653869731, 3.59032881975647066795230539223, 4.33411656556602806139700567678, 5.56568902393654669087316917983, 7.09215554927866160353667894650, 8.011129274506011085100366665662, 8.675988821449877719630208959943, 9.503141534336360059508005702329, 10.63512173247268900056372132772