| L(s) = 1 | + (−1.02 − 0.744i)2-s + (0.155 + 1.47i)3-s + (−0.122 − 0.376i)4-s + (1.90 + 3.29i)5-s + (0.940 − 1.62i)6-s + (−2.14 − 0.455i)7-s + (−0.937 + 2.88i)8-s + (0.779 − 0.165i)9-s + (0.503 − 4.78i)10-s + (0.636 − 0.706i)11-s + (0.536 − 0.238i)12-s + (−0.153 − 0.0683i)13-s + (1.85 + 2.06i)14-s + (−4.56 + 3.31i)15-s + (2.46 − 1.79i)16-s + (4.40 + 4.88i)17-s + ⋯ |
| L(s) = 1 | + (−0.724 − 0.526i)2-s + (0.0895 + 0.852i)3-s + (−0.0611 − 0.188i)4-s + (0.849 + 1.47i)5-s + (0.383 − 0.664i)6-s + (−0.809 − 0.172i)7-s + (−0.331 + 1.02i)8-s + (0.259 − 0.0552i)9-s + (0.159 − 1.51i)10-s + (0.191 − 0.212i)11-s + (0.154 − 0.0689i)12-s + (−0.0426 − 0.0189i)13-s + (0.495 + 0.550i)14-s + (−1.17 + 0.856i)15-s + (0.617 − 0.448i)16-s + (1.06 + 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.107 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.682904 + 0.760683i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.682904 + 0.760683i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + (1.02 + 0.744i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.155 - 1.47i)T + (-2.93 + 0.623i)T^{2} \) |
| 5 | \( 1 + (-1.90 - 3.29i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (2.14 + 0.455i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (-0.636 + 0.706i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (0.153 + 0.0683i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-4.40 - 4.88i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (1.05 - 0.468i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-1.43 + 4.40i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.08 - 0.785i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (1.93 - 3.35i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0343 - 0.326i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (8.79 - 3.91i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (4.56 - 3.31i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (7.17 - 1.52i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.277 - 2.63i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 - 1.74T + 61T^{2} \) |
| 67 | \( 1 + (0.276 + 0.478i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.11 - 0.236i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (-5.30 + 5.88i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (3.03 + 3.37i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.0341 + 0.324i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (-4.54 - 13.9i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-4.79 - 14.7i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.21911818674485199260781144702, −9.792655584838671834377764510479, −9.021988080181311446324305146440, −7.941915346476629860037292693058, −6.59933649530675869886547136425, −6.19248658596019054095438704119, −5.02507249269178565548064921050, −3.58878388090496656764246820805, −2.89748679667053715583552748646, −1.58917728540628980266929415502,
0.62773549440897589014276902432, 1.74669687757857756964913459129, 3.29227287616151721164130150673, 4.69639680120743087632441531726, 5.67653761934214983216531384017, 6.64142535832182638823276266689, 7.33128033103407794147139820169, 8.179830336130100755461709316379, 8.912389807634275402245580425382, 9.736647577605332903872945002683
