| 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 | \( 1 + (1.67 - 7.84i)T + (-88.4 - 39.7i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.75069386787736362223749370857, −10.08100722470743349525328970525, −9.435343586336350996806128947568, −8.099012940554851562965158923274, −7.50220949598917778561735167131, −6.79184313475631145553889575445, −5.26948586791160770911439055853, −4.58622623929930809906604809543, −3.56287609695316158560363001934, −1.44942318490895498146264575126,
1.00865754713979383021546709577, 2.30736410498210994252815403602, 3.66219818776785355327213115827, 4.86988297559823107637271435164, 5.86983682919163999223535340060, 7.32665333002017004620038189742, 7.970972749075844979998832021201, 8.793928122158753940493026242306, 9.785154564368910236982901260960, 10.65770825685838176955161212175
