| L(s) = 1 | + (−0.366 − 0.930i)2-s + (−0.0417 − 0.890i)3-s + (−0.731 + 0.681i)4-s + (−0.816 − 0.462i)5-s + (−0.813 + 0.365i)6-s + (2.09 + 1.86i)7-s + (0.902 + 0.430i)8-s + (2.19 − 0.206i)9-s + (−0.131 + 0.929i)10-s + (0.276 − 2.34i)11-s + (0.637 + 0.623i)12-s + (−2.04 − 0.288i)13-s + (0.967 − 2.63i)14-s + (−0.378 + 0.746i)15-s + (0.0702 − 0.997i)16-s + (7.35 − 0.866i)17-s + ⋯ |
| L(s) = 1 | + (−0.259 − 0.657i)2-s + (−0.0241 − 0.514i)3-s + (−0.365 + 0.340i)4-s + (−0.365 − 0.207i)5-s + (−0.332 + 0.149i)6-s + (0.793 + 0.705i)7-s + (0.319 + 0.152i)8-s + (0.731 − 0.0688i)9-s + (−0.0416 + 0.293i)10-s + (0.0832 − 0.707i)11-s + (0.184 + 0.179i)12-s + (−0.565 − 0.0801i)13-s + (0.258 − 0.704i)14-s + (−0.0976 + 0.192i)15-s + (0.0175 − 0.249i)16-s + (1.78 − 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00927 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00927 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.917334 - 0.925884i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.917334 - 0.925884i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.366 + 0.930i)T \) |
| 269 | \( 1 + (2.62 - 16.1i)T \) |
| good | 3 | \( 1 + (0.0417 + 0.890i)T + (-2.98 + 0.280i)T^{2} \) |
| 5 | \( 1 + (0.816 + 0.462i)T + (2.56 + 4.29i)T^{2} \) |
| 7 | \( 1 + (-2.09 - 1.86i)T + (0.818 + 6.95i)T^{2} \) |
| 11 | \( 1 + (-0.276 + 2.34i)T + (-10.6 - 2.55i)T^{2} \) |
| 13 | \( 1 + (2.04 + 0.288i)T + (12.4 + 3.60i)T^{2} \) |
| 17 | \( 1 + (-7.35 + 0.866i)T + (16.5 - 3.94i)T^{2} \) |
| 19 | \( 1 + (1.28 - 0.539i)T + (13.2 - 13.5i)T^{2} \) |
| 23 | \( 1 + (-2.66 + 0.125i)T + (22.8 - 2.15i)T^{2} \) |
| 29 | \( 1 + (-1.03 - 0.619i)T + (13.7 + 25.5i)T^{2} \) |
| 31 | \( 1 + (-0.195 + 0.818i)T + (-27.6 - 14.0i)T^{2} \) |
| 37 | \( 1 + (5.44 + 8.20i)T + (-14.3 + 34.0i)T^{2} \) |
| 41 | \( 1 + (-1.77 - 0.697i)T + (29.9 + 27.9i)T^{2} \) |
| 43 | \( 1 + (-1.29 + 2.16i)T + (-20.3 - 37.8i)T^{2} \) |
| 47 | \( 1 + (-1.24 - 0.782i)T + (20.2 + 42.4i)T^{2} \) |
| 53 | \( 1 + (3.42 - 10.8i)T + (-43.4 - 30.3i)T^{2} \) |
| 59 | \( 1 + (-0.661 - 0.709i)T + (-4.14 + 58.8i)T^{2} \) |
| 61 | \( 1 + (-0.909 + 4.24i)T + (-55.6 - 24.9i)T^{2} \) |
| 67 | \( 1 + (-3.96 - 3.69i)T + (4.70 + 66.8i)T^{2} \) |
| 71 | \( 1 + (1.25 - 0.459i)T + (54.1 - 45.9i)T^{2} \) |
| 73 | \( 1 + (-3.42 - 2.90i)T + (11.9 + 72.0i)T^{2} \) |
| 79 | \( 1 + (-2.12 - 8.06i)T + (-68.7 + 38.9i)T^{2} \) |
| 83 | \( 1 + (0.990 + 5.21i)T + (-77.2 + 30.4i)T^{2} \) |
| 89 | \( 1 + (-10.9 + 3.73i)T + (70.5 - 54.3i)T^{2} \) |
| 97 | \( 1 + (1.67 + 7.84i)T + (-88.4 + 39.7i)T^{2} \) |
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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
−10.65770825685838176955161212175, −9.785154564368910236982901260960, −8.793928122158753940493026242306, −7.970972749075844979998832021201, −7.32665333002017004620038189742, −5.86983682919163999223535340060, −4.86988297559823107637271435164, −3.66219818776785355327213115827, −2.30736410498210994252815403602, −1.00865754713979383021546709577,
1.44942318490895498146264575126, 3.56287609695316158560363001934, 4.58622623929930809906604809543, 5.26948586791160770911439055853, 6.79184313475631145553889575445, 7.50220949598917778561735167131, 8.099012940554851562965158923274, 9.435343586336350996806128947568, 10.08100722470743349525328970525, 10.75069386787736362223749370857
