| L(s) = 1 | + (−0.372 − 0.549i)2-s + (0.0541 − 0.998i)3-s + (0.576 − 1.44i)4-s + (2.31 − 1.07i)5-s + (−0.569 + 0.342i)6-s + (−0.243 + 0.876i)7-s + (−2.30 + 0.508i)8-s + (−0.994 − 0.108i)9-s + (−1.45 − 0.874i)10-s + (−3.02 + 2.86i)11-s + (−1.41 − 0.654i)12-s + (4.42 − 0.481i)13-s + (0.572 − 0.192i)14-s + (−0.944 − 2.37i)15-s + (−1.12 − 1.06i)16-s + (−1.03 − 3.74i)17-s + ⋯ |
| L(s) = 1 | + (−0.263 − 0.388i)2-s + (0.0312 − 0.576i)3-s + (0.288 − 0.724i)4-s + (1.03 − 0.479i)5-s + (−0.232 + 0.139i)6-s + (−0.0920 + 0.331i)7-s + (−0.816 + 0.179i)8-s + (−0.331 − 0.0360i)9-s + (−0.459 − 0.276i)10-s + (−0.912 + 0.864i)11-s + (−0.408 − 0.188i)12-s + (1.22 − 0.133i)13-s + (0.153 − 0.0515i)14-s + (−0.243 − 0.612i)15-s + (−0.280 − 0.266i)16-s + (−0.251 − 0.907i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0733 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0733 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.811714 - 0.873572i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.811714 - 0.873572i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.0541 + 0.998i)T \) |
| 59 | \( 1 + (-7.19 + 2.68i)T \) |
| good | 2 | \( 1 + (0.372 + 0.549i)T + (-0.740 + 1.85i)T^{2} \) |
| 5 | \( 1 + (-2.31 + 1.07i)T + (3.23 - 3.81i)T^{2} \) |
| 7 | \( 1 + (0.243 - 0.876i)T + (-5.99 - 3.60i)T^{2} \) |
| 11 | \( 1 + (3.02 - 2.86i)T + (0.595 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-4.42 + 0.481i)T + (12.6 - 2.79i)T^{2} \) |
| 17 | \( 1 + (1.03 + 3.74i)T + (-14.5 + 8.76i)T^{2} \) |
| 19 | \( 1 + (-2.39 + 1.82i)T + (5.08 - 18.3i)T^{2} \) |
| 23 | \( 1 + (3.29 - 6.20i)T + (-12.9 - 19.0i)T^{2} \) |
| 29 | \( 1 + (-0.798 + 1.17i)T + (-10.7 - 26.9i)T^{2} \) |
| 31 | \( 1 + (-8.51 - 6.47i)T + (8.29 + 29.8i)T^{2} \) |
| 37 | \( 1 + (3.02 + 0.664i)T + (33.5 + 15.5i)T^{2} \) |
| 41 | \( 1 + (0.0318 + 0.0600i)T + (-23.0 + 33.9i)T^{2} \) |
| 43 | \( 1 + (5.79 + 5.49i)T + (2.32 + 42.9i)T^{2} \) |
| 47 | \( 1 + (-8.63 - 3.99i)T + (30.4 + 35.8i)T^{2} \) |
| 53 | \( 1 + (1.86 - 1.12i)T + (24.8 - 46.8i)T^{2} \) |
| 61 | \( 1 + (0.597 + 0.881i)T + (-22.5 + 56.6i)T^{2} \) |
| 67 | \( 1 + (-0.687 + 0.151i)T + (60.8 - 28.1i)T^{2} \) |
| 71 | \( 1 + (6.98 + 3.23i)T + (45.9 + 54.1i)T^{2} \) |
| 73 | \( 1 + (8.75 - 2.94i)T + (58.1 - 44.1i)T^{2} \) |
| 79 | \( 1 + (-0.812 - 14.9i)T + (-78.5 + 8.54i)T^{2} \) |
| 83 | \( 1 + (0.914 + 5.57i)T + (-78.6 + 26.5i)T^{2} \) |
| 89 | \( 1 + (-1.96 + 2.89i)T + (-32.9 - 82.6i)T^{2} \) |
| 97 | \( 1 + (16.3 + 5.49i)T + (77.2 + 58.7i)T^{2} \) |
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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
−12.37104916769128741454127113768, −11.47071488242600251370003774902, −10.30895299871802599532527378997, −9.555717009804772454317079514686, −8.619426211823793857851737596530, −7.12401302225813314248950555493, −5.92144581068638881766111304443, −5.18366079259315788760124462267, −2.65939434333473142932642031594, −1.43074893750878251819776250232,
2.65101550322909078269900514575, 3.91708196591629966741051178615, 5.84853338762572651958408158013, 6.47561760633266460316640570793, 8.051156427745349858984693118839, 8.717393506908702669423446347350, 10.10835946001581969281115082236, 10.70166493014494753011395189918, 11.84940368376783832773416991924, 13.24960176764282867330188935878
