| L(s) = 1 | + 2.19e13·2-s + 9.03e20·3-s + 3.25e26·4-s − 2.93e30·5-s + 1.97e34·6-s + 7.30e36·7-s + 3.73e39·8-s + 4.93e41·9-s − 6.44e43·10-s − 1.23e45·11-s + 2.94e47·12-s + 1.56e48·13-s + 1.60e50·14-s − 2.65e51·15-s + 3.15e52·16-s + 4.26e53·17-s + 1.08e55·18-s − 1.72e55·19-s − 9.56e56·20-s + 6.60e57·21-s − 2.70e58·22-s + 1.74e59·23-s + 3.37e60·24-s + 2.18e60·25-s + 3.42e61·26-s + 1.53e62·27-s + 2.37e63·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 1.58·3-s + 2.10·4-s − 1.15·5-s + 2.79·6-s + 1.26·7-s + 1.94·8-s + 1.52·9-s − 2.03·10-s − 0.618·11-s + 3.34·12-s + 0.546·13-s + 2.22·14-s − 1.83·15-s + 1.31·16-s + 1.27·17-s + 2.68·18-s − 0.409·19-s − 2.43·20-s + 2.01·21-s − 1.08·22-s + 1.01·23-s + 3.08·24-s + 0.337·25-s + 0.962·26-s + 0.835·27-s + 2.66·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(88-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+87/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(44)\) |
\(\approx\) |
\(10.03261488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.03261488\) |
| \(L(\frac{89}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 2.19e13T + 1.54e26T^{2} \) |
| 3 | \( 1 - 9.03e20T + 3.23e41T^{2} \) |
| 5 | \( 1 + 2.93e30T + 6.46e60T^{2} \) |
| 7 | \( 1 - 7.30e36T + 3.33e73T^{2} \) |
| 11 | \( 1 + 1.23e45T + 3.99e90T^{2} \) |
| 13 | \( 1 - 1.56e48T + 8.18e96T^{2} \) |
| 17 | \( 1 - 4.26e53T + 1.11e107T^{2} \) |
| 19 | \( 1 + 1.72e55T + 1.78e111T^{2} \) |
| 23 | \( 1 - 1.74e59T + 2.95e118T^{2} \) |
| 29 | \( 1 - 1.40e63T + 1.69e127T^{2} \) |
| 31 | \( 1 + 6.33e64T + 5.60e129T^{2} \) |
| 37 | \( 1 + 2.97e68T + 2.71e136T^{2} \) |
| 41 | \( 1 + 4.83e69T + 2.05e140T^{2} \) |
| 43 | \( 1 - 1.18e71T + 1.29e142T^{2} \) |
| 47 | \( 1 + 2.96e72T + 2.96e145T^{2} \) |
| 53 | \( 1 + 1.89e75T + 1.02e150T^{2} \) |
| 59 | \( 1 + 2.41e76T + 1.15e154T^{2} \) |
| 61 | \( 1 - 4.87e77T + 2.10e155T^{2} \) |
| 67 | \( 1 + 1.66e79T + 7.38e158T^{2} \) |
| 71 | \( 1 - 5.91e80T + 1.14e161T^{2} \) |
| 73 | \( 1 + 3.16e80T + 1.28e162T^{2} \) |
| 79 | \( 1 - 9.23e81T + 1.24e165T^{2} \) |
| 83 | \( 1 + 2.59e83T + 9.11e166T^{2} \) |
| 89 | \( 1 + 1.03e84T + 3.95e169T^{2} \) |
| 97 | \( 1 - 1.01e85T + 7.06e172T^{2} \) |
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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
−14.82016323323506901778580724689, −13.98923135721901978307934666070, −12.55194619584887739719642366273, −11.09462002298286495113945405033, −8.315209156557834730227117742188, −7.39193947149212835344488009438, −5.06655878332264034460212201704, −3.87051340394422213480204515024, −3.05710998724344535873381840909, −1.69019304102728573214608690338,
1.69019304102728573214608690338, 3.05710998724344535873381840909, 3.87051340394422213480204515024, 5.06655878332264034460212201704, 7.39193947149212835344488009438, 8.315209156557834730227117742188, 11.09462002298286495113945405033, 12.55194619584887739719642366273, 13.98923135721901978307934666070, 14.82016323323506901778580724689
