| L(s) = 1 | + (−0.173 − 0.984i)2-s + (−2.04 + 1.71i)3-s + (−0.939 + 0.342i)4-s + (−3.54 − 1.28i)5-s + (2.04 + 1.71i)6-s + (0.168 − 0.292i)7-s + (0.5 + 0.866i)8-s + (0.713 − 4.04i)9-s + (−0.654 + 3.71i)10-s + (−0.5 − 0.866i)11-s + (1.33 − 2.30i)12-s + (0.595 + 0.499i)13-s + (−0.317 − 0.115i)14-s + (9.44 − 3.43i)15-s + (0.766 − 0.642i)16-s + (1.30 + 7.40i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−1.17 + 0.989i)3-s + (−0.469 + 0.171i)4-s + (−1.58 − 0.576i)5-s + (0.833 + 0.699i)6-s + (0.0638 − 0.110i)7-s + (0.176 + 0.306i)8-s + (0.237 − 1.34i)9-s + (−0.206 + 1.17i)10-s + (−0.150 − 0.261i)11-s + (0.384 − 0.666i)12-s + (0.165 + 0.138i)13-s + (−0.0848 − 0.0308i)14-s + (2.43 − 0.887i)15-s + (0.191 − 0.160i)16-s + (0.316 + 1.79i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.536748 - 0.0918763i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.536748 - 0.0918763i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-3.35 + 2.78i)T \) |
| good | 3 | \( 1 + (2.04 - 1.71i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (3.54 + 1.28i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.168 + 0.292i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (-0.595 - 0.499i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.30 - 7.40i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-5.12 + 1.86i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.43 + 8.12i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (1.33 - 2.30i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8.78T + 37T^{2} \) |
| 41 | \( 1 + (7.55 - 6.34i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-4.82 - 1.75i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.74 + 9.89i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (5.68 - 2.06i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (1.10 + 6.29i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (3.30 - 1.20i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.588 - 3.33i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (2.99 + 1.08i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-8.49 + 7.12i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-4.55 + 3.81i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.38 + 2.39i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-8.15 - 6.84i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (0.109 + 0.623i)T + (-91.1 + 33.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.14153869923991380492265411163, −10.62456684961704385994562522502, −9.567840227607740285075088595090, −8.531035682919933942777464704476, −7.70780506965261753141168428969, −6.17794476926834828072535785459, −4.93526526679643818243914725919, −4.29034363924329512759188385313, −3.42045950380619293585282511416, −0.73956709429801990933034152169,
0.77717349614431606365313259180, 3.23356852030092841160597748383, 4.76917435786833410071362133882, 5.64033917179446722926098146854, 6.94909078102900026590713278542, 7.27394018646780239556819726349, 7.968272985381712114791208643125, 9.301267003487547712338456070307, 10.70999249673483045058182797293, 11.41214920417271279803746254900
