L(s) = 1 | + (0.619 − 1.61i)3-s + (−0.590 − 1.02i)5-s + (−0.5 + 0.866i)7-s + (−2.23 − 2.00i)9-s + (−1.85 + 3.20i)11-s + (−0.5 − 0.866i)13-s + (−2.02 + 0.321i)15-s − 6.94·17-s − 1.94·19-s + (1.09 + 1.34i)21-s + (−2.80 − 4.85i)23-s + (1.80 − 3.12i)25-s + (−4.62 + 2.36i)27-s + (−0.119 + 0.207i)29-s + (0.830 + 1.43i)31-s + ⋯ |
L(s) = 1 | + (0.357 − 0.933i)3-s + (−0.264 − 0.457i)5-s + (−0.188 + 0.327i)7-s + (−0.744 − 0.668i)9-s + (−0.558 + 0.967i)11-s + (−0.138 − 0.240i)13-s + (−0.522 + 0.0830i)15-s − 1.68·17-s − 0.445·19-s + (0.238 + 0.293i)21-s + (−0.584 − 1.01i)23-s + (0.360 − 0.624i)25-s + (−0.890 + 0.455i)27-s + (−0.0222 + 0.0384i)29-s + (0.149 + 0.258i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3939775941\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3939775941\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.619 + 1.61i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.590 + 1.02i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.85 - 3.20i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 6.94T + 17T^{2} \) |
| 19 | \( 1 + 1.94T + 19T^{2} \) |
| 23 | \( 1 + (2.80 + 4.85i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.119 - 0.207i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.830 - 1.43i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 9.54T + 37T^{2} \) |
| 41 | \( 1 + (-5.09 - 8.81i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.11 + 1.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.91 + 5.04i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + (-1.30 - 2.25i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.80 + 6.58i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.75 - 3.03i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8.60T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + (-3.68 + 6.38i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.47 - 6.01i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 2.74T + 89T^{2} \) |
| 97 | \( 1 + (3.58 - 6.20i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.271941083441159342140071050895, −8.532838890843583645930011234554, −7.965514933090316221943893615943, −6.89175264745603749834164004305, −6.37291993468186173483183333616, −5.08933983818160914651623539277, −4.20494174126303139163850103735, −2.72532738441254203151386483255, −1.93874669782375860681415157286, −0.15615558218268396706815106302,
2.28103040173823964805981785659, 3.34937144359613095821118184064, 4.10224613653032528539750990557, 5.12905990001648044963211064024, 6.11697673985496305540929938962, 7.14479995095904556286821490490, 8.065007100529500665553084387946, 8.890754440887230268925805699500, 9.521892106730941461030068321830, 10.70321559572088518922746320205