| L(s) = 1 | + (0.220 − 0.184i)2-s + (−2.07 − 0.754i)3-s + (−0.332 + 1.88i)4-s + (0.608 + 3.45i)5-s + (−0.595 + 0.216i)6-s + (0.562 + 0.974i)8-s + (1.43 + 1.20i)9-s + (0.771 + 0.647i)10-s + (2.22 + 3.84i)11-s + (2.11 − 3.66i)12-s + (6.01 − 2.18i)13-s + (1.34 − 7.61i)15-s + (−3.29 − 1.20i)16-s + (−0.461 + 0.386i)17-s + 0.536·18-s + (1.45 + 4.10i)19-s + ⋯ |
| L(s) = 1 | + (0.155 − 0.130i)2-s + (−1.19 − 0.435i)3-s + (−0.166 + 0.944i)4-s + (0.272 + 1.54i)5-s + (−0.243 + 0.0885i)6-s + (0.198 + 0.344i)8-s + (0.476 + 0.400i)9-s + (0.244 + 0.204i)10-s + (0.669 + 1.15i)11-s + (0.610 − 1.05i)12-s + (1.66 − 0.606i)13-s + (0.346 − 1.96i)15-s + (−0.824 − 0.300i)16-s + (−0.111 + 0.0938i)17-s + 0.126·18-s + (0.333 + 0.942i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.457558 + 0.951337i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.457558 + 0.951337i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 + (-1.45 - 4.10i)T \) |
| good | 2 | \( 1 + (-0.220 + 0.184i)T + (0.347 - 1.96i)T^{2} \) |
| 3 | \( 1 + (2.07 + 0.754i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.608 - 3.45i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-2.22 - 3.84i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-6.01 + 2.18i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (0.461 - 0.386i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.281 + 1.59i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (1.58 + 1.33i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (2.47 - 4.29i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.232T + 37T^{2} \) |
| 41 | \( 1 + (5.66 + 2.06i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.93 - 10.9i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (6.91 + 5.80i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.677 + 3.84i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-4.35 + 3.65i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.918 + 5.21i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-6.92 - 5.81i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (0.546 + 3.09i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (3.04 + 1.10i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (4.88 + 1.77i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (2.01 - 3.48i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (5.20 - 1.89i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-10.7 + 8.99i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.62059465420650595888449095809, −9.806086252609351270798932893843, −8.546879276161164130781539282130, −7.58903698833210647343880143333, −6.72667871320039611132066865570, −6.35520634649849185476468398833, −5.28081733218803254731565328189, −3.91106796806810732186846288984, −3.16368742315081875373557675065, −1.71528140066717094766163944696,
0.61522569644571263246040720136, 1.44839571487543813873186111943, 3.88229770723414730687973433404, 4.67965993903805460431918499478, 5.55014390013206472656047171385, 5.89039588820885840944361311792, 6.75496210701941155359493253464, 8.518134840392703015250377877513, 8.984608103348076333216036206022, 9.694876012914646625953669042622
