L(s) = 1 | + (1.69 − 0.348i)3-s + (0.5 + 0.866i)5-s + (−1.08 + 2.41i)7-s + (2.75 − 1.18i)9-s + (1.17 + 0.675i)11-s + 4.94i·13-s + (1.15 + 1.29i)15-s + (2.87 − 4.97i)17-s + (2.84 − 1.64i)19-s + (−0.993 + 4.47i)21-s + (−4.33 + 2.50i)23-s + (−0.499 + 0.866i)25-s + (4.26 − 2.96i)27-s − 5.68i·29-s + (−2.45 − 1.41i)31-s + ⋯ |
L(s) = 1 | + (0.979 − 0.201i)3-s + (0.223 + 0.387i)5-s + (−0.409 + 0.912i)7-s + (0.918 − 0.394i)9-s + (0.353 + 0.203i)11-s + 1.37i·13-s + (0.297 + 0.334i)15-s + (0.696 − 1.20i)17-s + (0.653 − 0.377i)19-s + (−0.216 + 0.976i)21-s + (−0.903 + 0.521i)23-s + (−0.0999 + 0.173i)25-s + (0.820 − 0.571i)27-s − 1.05i·29-s + (−0.440 − 0.254i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.90211 + 0.417139i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.90211 + 0.417139i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.69 + 0.348i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (1.08 - 2.41i)T \) |
good | 11 | \( 1 + (-1.17 - 0.675i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4.94iT - 13T^{2} \) |
| 17 | \( 1 + (-2.87 + 4.97i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.84 + 1.64i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.33 - 2.50i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 5.68iT - 29T^{2} \) |
| 31 | \( 1 + (2.45 + 1.41i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.92 + 3.33i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3.73T + 41T^{2} \) |
| 43 | \( 1 - 4.06T + 43T^{2} \) |
| 47 | \( 1 + (2.84 + 4.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.26 - 0.730i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.34 - 7.52i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.65 + 0.954i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.51 - 4.36i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.38iT - 71T^{2} \) |
| 73 | \( 1 + (-14.1 - 8.16i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.41 + 4.18i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 + (8.08 + 13.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 12.0iT - 97T^{2} \) |
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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
−11.57049967683239618473218476086, −9.878708520269529664795046724642, −9.493943977346472064071827046075, −8.699129510537987759605897543702, −7.48631348197818731712084686246, −6.77252705426223268870258159436, −5.61789532050222591584695961825, −4.14408008463203194631867713191, −2.95967889714600831243729318901, −1.93016579008400675698332641412,
1.38338971620188910431068467300, 3.17668275280752308422342574141, 3.92241844442300222375718619344, 5.27021577079066714924897151923, 6.50832843635172370000150521511, 7.75340171052336542267799108022, 8.257589799409971328307467980679, 9.394739909854725833746658497891, 10.20087137047525077299336940217, 10.71810641736030740973594400342