| L(s) = 1 | + (−0.262 − 1.31i)3-s + (6.57 + 4.39i)5-s + (−6.01 + 4.01i)7-s + (6.64 − 2.75i)9-s + (9.31 + 1.85i)11-s + (−3.27 + 3.27i)13-s + (4.07 − 9.83i)15-s + (14.9 − 8.04i)17-s + (10.7 + 4.45i)19-s + (6.88 + 6.88i)21-s + (3.55 − 17.8i)23-s + (14.3 + 34.7i)25-s + (−12.0 − 18.1i)27-s + (−18.9 + 28.3i)29-s + (−53.0 + 10.5i)31-s + ⋯ |
| L(s) = 1 | + (−0.0874 − 0.439i)3-s + (1.31 + 0.879i)5-s + (−0.859 + 0.574i)7-s + (0.738 − 0.305i)9-s + (0.847 + 0.168i)11-s + (−0.251 + 0.251i)13-s + (0.271 − 0.655i)15-s + (0.880 − 0.473i)17-s + (0.565 + 0.234i)19-s + (0.327 + 0.327i)21-s + (0.154 − 0.777i)23-s + (0.575 + 1.39i)25-s + (−0.448 − 0.670i)27-s + (−0.653 + 0.977i)29-s + (−1.70 + 0.340i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.64842 + 0.198888i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64842 + 0.198888i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (-14.9 + 8.04i)T \) |
| good | 3 | \( 1 + (0.262 + 1.31i)T + (-8.31 + 3.44i)T^{2} \) |
| 5 | \( 1 + (-6.57 - 4.39i)T + (9.56 + 23.0i)T^{2} \) |
| 7 | \( 1 + (6.01 - 4.01i)T + (18.7 - 45.2i)T^{2} \) |
| 11 | \( 1 + (-9.31 - 1.85i)T + (111. + 46.3i)T^{2} \) |
| 13 | \( 1 + (3.27 - 3.27i)T - 169iT^{2} \) |
| 19 | \( 1 + (-10.7 - 4.45i)T + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (-3.55 + 17.8i)T + (-488. - 202. i)T^{2} \) |
| 29 | \( 1 + (18.9 - 28.3i)T + (-321. - 776. i)T^{2} \) |
| 31 | \( 1 + (53.0 - 10.5i)T + (887. - 367. i)T^{2} \) |
| 37 | \( 1 + (11.6 + 58.6i)T + (-1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (30.7 - 20.5i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-29.7 + 12.3i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (2.36 - 2.36i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (53.4 + 22.1i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (37.0 + 89.4i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (7.09 + 10.6i)T + (-1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 + 29.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-25.0 - 126. i)T + (-4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (35.7 + 23.8i)T + (2.03e3 + 4.92e3i)T^{2} \) |
| 79 | \( 1 + (-130. - 26.0i)T + (5.76e3 + 2.38e3i)T^{2} \) |
| 83 | \( 1 + (13.1 - 31.7i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-18.5 - 18.5i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-67.5 + 101. i)T + (-3.60e3 - 8.69e3i)T^{2} \) |
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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
−12.86310219195634088963893094294, −12.31149189493940352648443087320, −10.87266585781169249219193272059, −9.668345233681926509238700643173, −9.314770984514922204436790137383, −7.22592097870616802005563909579, −6.54269048730109213012651081242, −5.53216655563736577304409968689, −3.36934652713670253804653957742, −1.84124784951826316548211714156,
1.45371612265136412878062455546, 3.68040408968803817179165295141, 5.11905549577845883063177004826, 6.16020976781721870886104204702, 7.52162906493665273771700670038, 9.251990169582956281832619652100, 9.681131602173730572414271911181, 10.55987722529354227725159185749, 12.10864781232410441353449246160, 13.19927156236327501942924982964
