L(s) = 1 | + (−1 + i)3-s + (−2 − 2i)5-s + i·7-s + i·9-s + (2 + 2i)11-s + (−4 + 4i)13-s + 4·15-s − 2·17-s + (−5 + 5i)19-s + (−1 − i)21-s − 4i·23-s + 3i·25-s + (−4 − 4i)27-s + (1 − i)29-s + 10·31-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.577i)3-s + (−0.894 − 0.894i)5-s + 0.377i·7-s + 0.333i·9-s + (0.603 + 0.603i)11-s + (−1.10 + 1.10i)13-s + 1.03·15-s − 0.485·17-s + (−1.14 + 1.14i)19-s + (−0.218 − 0.218i)21-s − 0.834i·23-s + 0.600i·25-s + (−0.769 − 0.769i)27-s + (0.185 − 0.185i)29-s + 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + (1 - i)T - 3iT^{2} \) |
| 5 | \( 1 + (2 + 2i)T + 5iT^{2} \) |
| 11 | \( 1 + (-2 - 2i)T + 11iT^{2} \) |
| 13 | \( 1 + (4 - 4i)T - 13iT^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + (5 - 5i)T - 19iT^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + (-1 + i)T - 29iT^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 + (-5 - 5i)T + 37iT^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 + (-6 - 6i)T + 43iT^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + (5 + 5i)T + 53iT^{2} \) |
| 59 | \( 1 + (7 + 7i)T + 59iT^{2} \) |
| 61 | \( 1 + (6 - 6i)T - 61iT^{2} \) |
| 67 | \( 1 + (4 - 4i)T - 67iT^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + (-5 + 5i)T - 83iT^{2} \) |
| 89 | \( 1 - 14iT - 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.139370001345980821841055281661, −7.930406657015532604771641967739, −6.61485132605948040018551901437, −6.19972666765522919483603407524, −4.86266395508102993342833581771, −4.50773837132717486845064043987, −4.20242190913211487620605682386, −2.64344970701241355189009337712, −1.61538978179739499130008833875, 0,
0.865066224765272365451958996456, 2.46581856000856014163522815891, 3.27343777741381817746288983069, 4.10746844638981451902451689388, 4.99508172689265751691000283679, 6.12674548002353674687671357021, 6.53842449428804265647958744579, 7.38800309676331743424737118713, 7.66289703525288581899893429430