| L(s) = 1 | + (−2.30 − 1.54i)3-s + (5.10 − 1.01i)5-s + (−1.70 − 0.339i)7-s + (−0.502 − 1.21i)9-s + (−7.03 − 10.5i)11-s + (7.79 − 7.79i)13-s + (−13.3 − 5.51i)15-s + (−3.27 − 16.6i)17-s + (5.87 − 14.1i)19-s + (3.40 + 3.40i)21-s + (5.68 − 3.80i)23-s + (1.88 − 0.780i)25-s + (−5.57 + 28.0i)27-s + (8.05 + 40.4i)29-s + (2.61 − 3.90i)31-s + ⋯ |
| L(s) = 1 | + (−0.768 − 0.513i)3-s + (1.02 − 0.202i)5-s + (−0.243 − 0.0484i)7-s + (−0.0557 − 0.134i)9-s + (−0.639 − 0.956i)11-s + (0.599 − 0.599i)13-s + (−0.888 − 0.367i)15-s + (−0.192 − 0.981i)17-s + (0.309 − 0.746i)19-s + (0.162 + 0.162i)21-s + (0.247 − 0.165i)23-s + (0.0753 − 0.0312i)25-s + (−0.206 + 1.03i)27-s + (0.277 + 1.39i)29-s + (0.0842 − 0.126i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0465 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0465 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.775623 - 0.812608i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.775623 - 0.812608i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (3.27 + 16.6i)T \) |
| good | 3 | \( 1 + (2.30 + 1.54i)T + (3.44 + 8.31i)T^{2} \) |
| 5 | \( 1 + (-5.10 + 1.01i)T + (23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (1.70 + 0.339i)T + (45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (7.03 + 10.5i)T + (-46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (-7.79 + 7.79i)T - 169iT^{2} \) |
| 19 | \( 1 + (-5.87 + 14.1i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-5.68 + 3.80i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (-8.05 - 40.4i)T + (-776. + 321. i)T^{2} \) |
| 31 | \( 1 + (-2.61 + 3.90i)T + (-367. - 887. i)T^{2} \) |
| 37 | \( 1 + (-22.6 - 15.1i)T + (523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-28.9 - 5.74i)T + (1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (-13.6 - 32.9i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-13.6 + 13.6i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-16.1 + 38.8i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-68.7 + 28.4i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (8.57 - 43.0i)T + (-3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 - 120. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-35.3 - 23.6i)T + (1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (39.8 - 7.93i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (60.9 + 91.2i)T + (-2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (-130. - 53.8i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (89.1 + 89.1i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-24.4 - 122. i)T + (-8.69e3 + 3.60e3i)T^{2} \) |
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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
−13.00438879794751222964886074785, −11.62756928349824214437802258128, −10.83509305661767743362996239627, −9.631669288486269660539735834737, −8.578980015223480590614113955332, −7.03181612650652047680604826886, −5.98139275732712261180783598655, −5.22563912194056157882220269019, −2.96553344224782611043874442378, −0.849853701140885005811845983027,
2.14493834970163204516414974588, 4.25604891384691207837757078051, 5.58754386201953657118389831314, 6.34322974741615095367817671533, 7.905303951647440582787571461871, 9.445722096784792523145506636689, 10.21342721158191556125729564438, 10.98136432118495643242374665200, 12.19217418865162244712179642219, 13.26179740157366054353338672705
