L(s) = 1 | + (0.468 + 1.66i)3-s + 3.02i·5-s + 3.62i·7-s + (−2.56 + 1.56i)9-s − 3.33·11-s + 1.32·13-s + (−5.03 + 1.41i)15-s + 7.12i·17-s − 5.20i·19-s + (−6.04 + 1.69i)21-s + 4.41·23-s − 4.12·25-s + (−3.80 − 3.54i)27-s − 6.41i·29-s − 0.794i·31-s + ⋯ |
L(s) = 1 | + (0.270 + 0.962i)3-s + 1.35i·5-s + 1.36i·7-s + (−0.853 + 0.520i)9-s − 1.00·11-s + 0.367·13-s + (−1.30 + 0.365i)15-s + 1.72i·17-s − 1.19i·19-s + (−1.31 + 0.370i)21-s + 0.920·23-s − 0.824·25-s + (−0.731 − 0.681i)27-s − 1.19i·29-s − 0.142i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.455902194\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.455902194\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.468 - 1.66i)T \) |
good | 5 | \( 1 - 3.02iT - 5T^{2} \) |
| 7 | \( 1 - 3.62iT - 7T^{2} \) |
| 11 | \( 1 + 3.33T + 11T^{2} \) |
| 13 | \( 1 - 1.32T + 13T^{2} \) |
| 17 | \( 1 - 7.12iT - 17T^{2} \) |
| 19 | \( 1 + 5.20iT - 19T^{2} \) |
| 23 | \( 1 - 4.41T + 23T^{2} \) |
| 29 | \( 1 + 6.41iT - 29T^{2} \) |
| 31 | \( 1 + 0.794iT - 31T^{2} \) |
| 37 | \( 1 - 8.10T + 37T^{2} \) |
| 41 | \( 1 - 4iT - 41T^{2} \) |
| 43 | \( 1 - 1.46iT - 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 - 3.76iT - 53T^{2} \) |
| 59 | \( 1 + 5.73T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 7.08iT - 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 4.86iT - 79T^{2} \) |
| 83 | \( 1 - 0.410T + 83T^{2} \) |
| 89 | \( 1 - 4.87iT - 89T^{2} \) |
| 97 | \( 1 - 1.12T + 97T^{2} \) |
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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
−9.895231075836523388772744348235, −9.153723170078416800854508963086, −8.372133439972329329036624365430, −7.69072255856444150754946037787, −6.40582315407733859175121327844, −5.85936271537461284115617605669, −4.93748189477817703246920600889, −3.83615829475051786029624806930, −2.75906390340483858348717149787, −2.43807067415080929251834228558,
0.58103646907447817740982229299, 1.32430756383840911539612711523, 2.76293076595523426218008613761, 3.88988095140659182818568480894, 4.96082324541542417153836949813, 5.60010257080055007540951460674, 6.89206896871466701826782052632, 7.45065913655568624134061529751, 8.139878979052528794363501186141, 8.874192214310794368049010221099