| L(s) = 1 | + (0.173 − 0.984i)2-s + (−1.06 − 1.36i)3-s + (−0.939 − 0.342i)4-s + (−2.37 + 1.99i)5-s + (−1.52 + 0.811i)6-s + (0.939 − 0.342i)7-s + (−0.5 + 0.866i)8-s + (−0.730 + 2.90i)9-s + (1.55 + 2.68i)10-s + (−2.26 − 1.90i)11-s + (0.533 + 1.64i)12-s + (0.665 + 3.77i)13-s + (−0.173 − 0.984i)14-s + (5.25 + 1.12i)15-s + (0.766 + 0.642i)16-s + (2.94 + 5.10i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.615 − 0.788i)3-s + (−0.469 − 0.171i)4-s + (−1.06 + 0.892i)5-s + (−0.624 + 0.331i)6-s + (0.355 − 0.129i)7-s + (−0.176 + 0.306i)8-s + (−0.243 + 0.969i)9-s + (0.490 + 0.849i)10-s + (−0.682 − 0.572i)11-s + (0.154 + 0.475i)12-s + (0.184 + 1.04i)13-s + (−0.0464 − 0.263i)14-s + (1.35 + 0.289i)15-s + (0.191 + 0.160i)16-s + (0.714 + 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.392185 + 0.228821i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.392185 + 0.228821i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 3 | \( 1 + (1.06 + 1.36i)T \) |
| 7 | \( 1 + (-0.939 + 0.342i)T \) |
| good | 5 | \( 1 + (2.37 - 1.99i)T + (0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (2.26 + 1.90i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.665 - 3.77i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-2.94 - 5.10i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.24 - 2.14i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.13 + 2.23i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.954 - 5.41i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-4.28 - 1.55i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (1.07 + 1.86i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.307 + 1.74i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (4.28 + 3.59i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (9.96 - 3.62i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 8.97T + 53T^{2} \) |
| 59 | \( 1 + (6.15 - 5.16i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (2.47 - 0.900i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.134 - 0.763i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-7.91 - 13.7i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.03 - 10.4i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.137 + 0.781i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.19 + 6.79i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-7.84 + 13.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.25 - 6.09i)T + (16.8 + 95.5i)T^{2} \) |
| 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
−11.54423834982416948092864238873, −10.80874297109037001069061846185, −10.24745292858216214004068183956, −8.420551332794365307164582095981, −7.88633860807459949939621701050, −6.76586299785488915043356452900, −5.79760401225931641304872018544, −4.39347772572031239917252227196, −3.27675307627935948980703331178, −1.74843942381607349601713479724,
0.31791134773541041249302313483, 3.38557517509230401295619988478, 4.68134433407866993109915905199, 5.04040343360968448186336516539, 6.22236591206022817341663525389, 7.76506591415045752807936859812, 8.115556153682050516899503648998, 9.395361227459377255218006141613, 10.14968386678844722389343286027, 11.40991059289385454139294445144
