L(s) = 1 | + (0.261 − 1.48i)3-s + (0.766 + 0.642i)5-s + (1.67 + 2.90i)7-s + (0.680 + 0.247i)9-s + (1.23 − 2.14i)11-s + (0.324 + 1.84i)13-s + (1.15 − 0.969i)15-s + (0.341 − 0.124i)17-s + (−3.65 + 2.36i)19-s + (4.75 − 1.72i)21-s + (5.71 − 4.79i)23-s + (0.173 + 0.984i)25-s + (2.80 − 4.86i)27-s + (−3.04 − 1.10i)29-s + (−3.78 − 6.55i)31-s + ⋯ |
L(s) = 1 | + (0.151 − 0.857i)3-s + (0.342 + 0.287i)5-s + (0.633 + 1.09i)7-s + (0.226 + 0.0825i)9-s + (0.373 − 0.646i)11-s + (0.0900 + 0.510i)13-s + (0.298 − 0.250i)15-s + (0.0829 − 0.0301i)17-s + (−0.839 + 0.543i)19-s + (1.03 − 0.377i)21-s + (1.19 − 1.00i)23-s + (0.0347 + 0.196i)25-s + (0.540 − 0.936i)27-s + (−0.565 − 0.205i)29-s + (−0.679 − 1.17i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.64731 - 0.214725i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.64731 - 0.214725i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 + (3.65 - 2.36i)T \) |
good | 3 | \( 1 + (-0.261 + 1.48i)T + (-2.81 - 1.02i)T^{2} \) |
| 7 | \( 1 + (-1.67 - 2.90i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.23 + 2.14i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.324 - 1.84i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.341 + 0.124i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-5.71 + 4.79i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (3.04 + 1.10i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (3.78 + 6.55i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 3.34T + 37T^{2} \) |
| 41 | \( 1 + (1.49 - 8.45i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (8.07 + 6.77i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-11.0 - 4.02i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (2.01 - 1.69i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (11.3 - 4.11i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (8.67 - 7.28i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (2.31 + 0.840i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (0.0802 + 0.0673i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (1.34 - 7.63i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.948 + 5.37i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.462 - 0.800i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.38 + 7.86i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (6.64 - 2.41i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.43025789489781580228945757085, −10.55490123438056780655473474398, −9.247348542255357801314618302376, −8.537953965494021523284550564396, −7.58840963809591501512221636131, −6.50992615212335252155735546548, −5.75650891320445862188645956377, −4.38626079108422046275636511663, −2.66134902135247703153820156007, −1.62071034802330350571774624376,
1.47465051502578330026075542080, 3.46390247563468659687574981667, 4.46808907468825193822655886950, 5.20858811161440117172657343345, 6.79547352984897381194925316264, 7.60093811459637835940664079922, 8.893261121788960879978500356981, 9.563778396733456105213448095912, 10.58373684611796748095657197191, 10.96507370214139674948934950077