| L(s) = 1 | + (4.5 − 7.79i)3-s + (−32.7 − 56.7i)5-s + (−40.6 − 123. i)7-s + (−40.5 − 70.1i)9-s + (−122. + 212. i)11-s + 434.·13-s − 590.·15-s + (551. − 954. i)17-s + (−1.43e3 − 2.49e3i)19-s + (−1.14e3 − 237. i)21-s + (−2.11e3 − 3.66e3i)23-s + (−587. + 1.01e3i)25-s − 729·27-s + 4.96e3·29-s + (−4.39e3 + 7.60e3i)31-s + ⋯ |
| L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.586 − 1.01i)5-s + (−0.313 − 0.949i)7-s + (−0.166 − 0.288i)9-s + (−0.305 + 0.529i)11-s + 0.713·13-s − 0.677·15-s + (0.462 − 0.801i)17-s + (−0.914 − 1.58i)19-s + (−0.565 − 0.117i)21-s + (−0.833 − 1.44i)23-s + (−0.187 + 0.325i)25-s − 0.192·27-s + 1.09·29-s + (−0.820 + 1.42i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8987096871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8987096871\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.5 + 7.79i)T \) |
| 7 | \( 1 + (40.6 + 123. i)T \) |
| good | 5 | \( 1 + (32.7 + 56.7i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (122. - 212. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 - 434.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-551. + 954. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (1.43e3 + 2.49e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (2.11e3 + 3.66e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 - 4.96e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (4.39e3 - 7.60e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-1.22e3 - 2.11e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + 3.66e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.19e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.63e3 - 2.83e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.51e3 + 2.61e3i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (2.57e4 - 4.45e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-6.65e3 - 1.15e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.54e4 + 2.67e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 4.18e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (1.73e4 - 2.99e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-3.87e4 - 6.71e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + 1.00e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + (2.03e4 + 3.53e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.40e5T + 8.58e9T^{2} \) |
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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
−10.17280402876628905283259666346, −8.938535003942496739267139182125, −8.326608439416653128220680522322, −7.28171789407181508001249805906, −6.49971569999435894891546937498, −4.88642637732233273015382290831, −4.13742844870521718234176784084, −2.72119513200681948576377328553, −1.10015129013693860208927195640, −0.24671663315313065664394785876,
1.98325521235115412132231605525, 3.30871963207071175035395504726, 3.87591368335653290594814474029, 5.66524850394053160899497326323, 6.27022547123651092358562959191, 7.79573323556952346675004793094, 8.354231335894596811151225647118, 9.531390322965518325499882400064, 10.43440422303049655290567575630, 11.17505037092717498926217695595
