| L(s) = 1 | + (−1.41 − 0.660i)3-s + (−0.949 − 2.02i)5-s + (−1.12 + 4.18i)7-s + (−0.359 − 0.428i)9-s + (−0.627 + 1.08i)11-s + (1.92 + 4.12i)13-s + (0.00842 + 3.49i)15-s + (0.0727 + 0.831i)17-s + (3.83 + 2.06i)19-s + (4.35 − 5.18i)21-s + (0.587 + 0.411i)23-s + (−3.19 + 3.84i)25-s + (1.43 + 5.37i)27-s + (−3.22 + 2.70i)29-s + (−6.47 + 3.74i)31-s + ⋯ |
| L(s) = 1 | + (−0.817 − 0.381i)3-s + (−0.424 − 0.905i)5-s + (−0.423 + 1.58i)7-s + (−0.119 − 0.142i)9-s + (−0.189 + 0.327i)11-s + (0.533 + 1.14i)13-s + (0.00217 + 0.902i)15-s + (0.0176 + 0.201i)17-s + (0.880 + 0.474i)19-s + (0.949 − 1.13i)21-s + (0.122 + 0.0858i)23-s + (−0.639 + 0.769i)25-s + (0.276 + 1.03i)27-s + (−0.599 + 0.503i)29-s + (−1.16 + 0.671i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0247 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0247 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.406824 + 0.396869i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.406824 + 0.396869i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.949 + 2.02i)T \) |
| 19 | \( 1 + (-3.83 - 2.06i)T \) |
| good | 3 | \( 1 + (1.41 + 0.660i)T + (1.92 + 2.29i)T^{2} \) |
| 7 | \( 1 + (1.12 - 4.18i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (0.627 - 1.08i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.92 - 4.12i)T + (-8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (-0.0727 - 0.831i)T + (-16.7 + 2.95i)T^{2} \) |
| 23 | \( 1 + (-0.587 - 0.411i)T + (7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (3.22 - 2.70i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (6.47 - 3.74i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.31 + 1.31i)T - 37iT^{2} \) |
| 41 | \( 1 + (2.83 + 7.79i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-6.72 - 9.60i)T + (-14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (9.51 + 0.832i)T + (46.2 + 8.16i)T^{2} \) |
| 53 | \( 1 + (5.49 - 7.84i)T + (-18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (2.93 + 2.46i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.31 + 7.44i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.616 + 7.04i)T + (-65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-11.6 - 2.05i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.19 + 2.56i)T + (-46.9 - 55.9i)T^{2} \) |
| 79 | \( 1 + (5.16 - 1.88i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-7.22 - 1.93i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-3.31 - 1.20i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (5.44 - 0.476i)T + (95.5 - 16.8i)T^{2} \) |
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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
−11.71001379684195367456355631750, −11.07263057681717519689608476021, −9.292819316786769734049419746483, −9.122104888749233139716004157764, −7.897045709866135390167557081913, −6.62992482882776763498971102651, −5.73574965020482228406852108820, −5.02472508342387490630440568841, −3.48038157984662997749987226289, −1.66735834279711901428139891436,
0.42494364535310490103427595573, 3.08639896389127500819604279511, 4.00684613659003188142131947993, 5.31581392332424626603938860914, 6.34495174473809424184780666431, 7.34876647548678127255377119536, 8.034328281869486620205855763364, 9.720807911167160136962838485458, 10.47011898415342794997875118100, 11.02332731720094441070237244725
