| L(s) = 1 | + (0.322 − 0.0684i)2-s + (−0.519 + 1.65i)3-s + (−1.72 + 0.769i)4-s + (−1.71 − 1.42i)5-s + (−0.0543 + 0.567i)6-s + (−1.22 − 2.11i)7-s + (−1.03 + 0.753i)8-s + (−2.45 − 1.71i)9-s + (−0.651 − 0.342i)10-s + (−1.65 + 0.351i)11-s + (−0.372 − 3.25i)12-s + (2.11 + 0.450i)13-s + (−0.537 − 0.597i)14-s + (3.25 − 2.09i)15-s + (2.24 − 2.49i)16-s + (−3.09 + 2.24i)17-s + ⋯ |
| L(s) = 1 | + (0.227 − 0.0484i)2-s + (−0.300 + 0.953i)3-s + (−0.863 + 0.384i)4-s + (−0.769 − 0.639i)5-s + (−0.0221 + 0.231i)6-s + (−0.461 − 0.798i)7-s + (−0.366 + 0.266i)8-s + (−0.819 − 0.572i)9-s + (−0.206 − 0.108i)10-s + (−0.498 + 0.106i)11-s + (−0.107 − 0.939i)12-s + (0.587 + 0.124i)13-s + (−0.143 − 0.159i)14-s + (0.840 − 0.541i)15-s + (0.562 − 0.624i)16-s + (−0.750 + 0.545i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00723819 - 0.0340541i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00723819 - 0.0340541i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.519 - 1.65i)T \) |
| 5 | \( 1 + (1.71 + 1.42i)T \) |
| good | 2 | \( 1 + (-0.322 + 0.0684i)T + (1.82 - 0.813i)T^{2} \) |
| 7 | \( 1 + (1.22 + 2.11i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.65 - 0.351i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-2.11 - 0.450i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (3.09 - 2.24i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (4.15 - 3.01i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (5.83 + 6.48i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (0.606 - 5.76i)T + (-28.3 - 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.743 - 7.07i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-2.56 - 7.88i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (4.23 + 0.900i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (3.28 + 5.69i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.832 + 7.92i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (2.04 + 1.48i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.942 - 0.200i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-8.52 + 1.81i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (0.353 + 3.36i)T + (-65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (0.529 + 0.384i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.37 + 10.3i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (0.658 - 6.26i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (13.0 + 5.81i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (2.96 - 9.11i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.161 + 1.53i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−12.70497403521815089436355339220, −11.98574991713354444905242840578, −10.69881575710821196727849136445, −10.05010466509436454773205705343, −8.634141537642971402051043725303, −8.359838783797533025119918151225, −6.58893993766464976630098380554, −5.11516728590048004766846801006, −4.20143839237050562034486516178, −3.56412046273451583336705373170,
0.02727654911378826868028215214, 2.57624268325021739774646053411, 4.15427304783668529918640171791, 5.67705909520173703996436800310, 6.38112585296164464573761586176, 7.70156915154385513654956787058, 8.571680281173967475072687414348, 9.700302107645781328165228632617, 11.04730983545102213302974790598, 11.71386910469071921965016540066
