| L(s) = 1 | + (0.671 + 0.244i)3-s + (−0.173 − 0.984i)5-s + (1.15 − 2.00i)7-s + (−1.90 − 1.60i)9-s + (−1.57 − 2.72i)11-s + (5.20 − 1.89i)13-s + (0.124 − 0.703i)15-s + (−2.31 + 1.94i)17-s + (4.24 + 0.973i)19-s + (1.26 − 1.06i)21-s + (−0.283 + 1.60i)23-s + (−0.939 + 0.342i)25-s + (−1.96 − 3.39i)27-s + (3.91 + 3.28i)29-s + (1.01 − 1.75i)31-s + ⋯ |
| L(s) = 1 | + (0.387 + 0.141i)3-s + (−0.0776 − 0.440i)5-s + (0.438 − 0.759i)7-s + (−0.635 − 0.533i)9-s + (−0.474 − 0.822i)11-s + (1.44 − 0.525i)13-s + (0.0320 − 0.181i)15-s + (−0.561 + 0.471i)17-s + (0.974 + 0.223i)19-s + (0.276 − 0.232i)21-s + (−0.0591 + 0.335i)23-s + (−0.187 + 0.0684i)25-s + (−0.377 − 0.653i)27-s + (0.726 + 0.609i)29-s + (0.181 − 0.314i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33533 - 0.677101i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33533 - 0.677101i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (-4.24 - 0.973i)T \) |
| good | 3 | \( 1 + (-0.671 - 0.244i)T + (2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (-1.15 + 2.00i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.57 + 2.72i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-5.20 + 1.89i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (2.31 - 1.94i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.283 - 1.60i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.91 - 3.28i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.01 + 1.75i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.94T + 37T^{2} \) |
| 41 | \( 1 + (-6.75 - 2.45i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.0418 + 0.237i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (5.49 + 4.60i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (0.521 - 2.95i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (0.466 - 0.391i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.87 - 10.6i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (11.5 + 9.65i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.791 - 4.49i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-13.6 - 4.95i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (4.11 + 1.49i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (2.97 - 5.15i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-8.52 + 3.10i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-7.30 + 6.12i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.11280781362393132875018726138, −10.46398114994477316105659071996, −9.183401256495883831131360615655, −8.431984938338288085665980183629, −7.77929932164200700941459891032, −6.32424499192711113483015546248, −5.41265301213260728676062832727, −4.01680674861921510573139542152, −3.11927657673555234949755576098, −1.06175585440897607432692553074,
2.01164617801130439054672512185, 3.07517621199499142541106656353, 4.61457142181316106465380873164, 5.69217527976818043812004957282, 6.81555650393411654335773187657, 7.906896275033535263304376883376, 8.638401943163074509707089255804, 9.526679838614869705909684074688, 10.77750134569505701456422569955, 11.40332769511884625886332951033
