| L(s) = 1 | + (1.22 − 0.702i)2-s + (0.106 + 1.22i)3-s + (1.01 − 1.72i)4-s + (−2.11 + 0.727i)5-s + (0.989 + 1.42i)6-s + (1.61 + 0.431i)7-s + (0.0324 − 2.82i)8-s + (1.47 − 0.259i)9-s + (−2.08 + 2.37i)10-s + (5.01 + 2.89i)11-s + (2.21 + 1.05i)12-s + (−0.0482 + 0.551i)13-s + (2.27 − 0.601i)14-s + (−1.11 − 2.50i)15-s + (−1.94 − 3.49i)16-s + (3.11 − 2.18i)17-s + ⋯ |
| L(s) = 1 | + (0.867 − 0.496i)2-s + (0.0617 + 0.705i)3-s + (0.506 − 0.862i)4-s + (−0.945 + 0.325i)5-s + (0.403 + 0.581i)6-s + (0.608 + 0.163i)7-s + (0.0114 − 0.999i)8-s + (0.490 − 0.0865i)9-s + (−0.659 + 0.751i)10-s + (1.51 + 0.873i)11-s + (0.639 + 0.304i)12-s + (−0.0133 + 0.152i)13-s + (0.609 − 0.160i)14-s + (−0.287 − 0.647i)15-s + (−0.486 − 0.873i)16-s + (0.756 − 0.529i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.22900 - 0.0629705i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.22900 - 0.0629705i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 + 0.702i)T \) |
| 5 | \( 1 + (2.11 - 0.727i)T \) |
| 19 | \( 1 + (2.94 - 3.20i)T \) |
| good | 3 | \( 1 + (-0.106 - 1.22i)T + (-2.95 + 0.520i)T^{2} \) |
| 7 | \( 1 + (-1.61 - 0.431i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-5.01 - 2.89i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.0482 - 0.551i)T + (-12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (-3.11 + 2.18i)T + (5.81 - 15.9i)T^{2} \) |
| 23 | \( 1 + (2.98 + 6.40i)T + (-14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (5.77 - 1.01i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (6.23 - 3.60i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.800 - 0.800i)T + 37iT^{2} \) |
| 41 | \( 1 + (-0.851 - 0.714i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (9.66 + 4.50i)T + (27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (0.187 + 0.131i)T + (16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (-11.6 + 5.42i)T + (34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (0.103 - 0.587i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (9.10 + 3.31i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-3.42 - 2.39i)T + (22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (-3.02 - 8.31i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-1.23 + 0.108i)T + (71.8 - 12.6i)T^{2} \) |
| 79 | \( 1 + (2.51 + 2.10i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.67 + 9.98i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (3.19 + 3.81i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-4.73 + 3.31i)T + (33.1 - 91.1i)T^{2} \) |
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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
−11.53514943251957582430279548680, −10.54255766855966427134950339689, −9.844232336646904918047663652909, −8.764863862036712710462590649083, −7.34454376965331901551677149461, −6.55931323398944770742436513210, −5.05171346840230881517190288083, −4.15977863037529970699727775704, −3.60208122439269185403088714930, −1.76805574530253446370683516841,
1.56835560387591530865884908827, 3.58411684149657798851787112334, 4.25732699208205013686848049857, 5.60016278231458848405580637516, 6.66686588046433140072989269430, 7.58095242481370329562548875767, 8.130868312155669398798621251340, 9.200949505018582608583212381928, 11.00161188346462518764543952525, 11.61379225456698876402596429667
