L(s) = 1 | + (2.89 + 0.795i)3-s + (−0.355 + 0.615i)5-s + (−2.70 + 1.56i)7-s + (7.73 + 4.60i)9-s + (14.3 − 8.30i)11-s + (9.17 − 15.8i)13-s + (−1.51 + 1.49i)15-s − 9.69·17-s − 8.20i·19-s + (−9.06 + 2.36i)21-s + (−1.94 − 1.12i)23-s + (12.2 + 21.2i)25-s + (18.7 + 19.4i)27-s + (20.8 + 36.0i)29-s + (21.6 + 12.4i)31-s + ⋯ |
L(s) = 1 | + (0.964 + 0.265i)3-s + (−0.0710 + 0.123i)5-s + (−0.386 + 0.223i)7-s + (0.859 + 0.511i)9-s + (1.30 − 0.754i)11-s + (0.705 − 1.22i)13-s + (−0.101 + 0.0998i)15-s − 0.570·17-s − 0.431i·19-s + (−0.431 + 0.112i)21-s + (−0.0847 − 0.0489i)23-s + (0.489 + 0.848i)25-s + (0.692 + 0.721i)27-s + (0.717 + 1.24i)29-s + (0.697 + 0.402i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.693646438\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.693646438\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.89 - 0.795i)T \) |
good | 5 | \( 1 + (0.355 - 0.615i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (2.70 - 1.56i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-14.3 + 8.30i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-9.17 + 15.8i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 9.69T + 289T^{2} \) |
| 19 | \( 1 + 8.20iT - 361T^{2} \) |
| 23 | \( 1 + (1.94 + 1.12i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-20.8 - 36.0i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-21.6 - 12.4i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 40.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-25.6 + 44.5i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (56.6 - 32.7i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (29.2 - 16.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 90.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (66.2 + 38.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (1.35 + 2.35i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (34.5 + 19.9i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 38.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + (94.4 - 54.5i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-113. + 65.5i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 38.0T + 7.92e3T^{2} \) |
| 97 | \( 1 + (12.1 + 21.1i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.54096718305284716357540509224, −9.435300503872681361919681157518, −8.826137161414723497879224317245, −8.128925833986312311346588585276, −6.95840160965567332533817050948, −6.10227237533229624590825873460, −4.76180339469916852714857344988, −3.54151847006396938863928587243, −2.92498079042197092582354866077, −1.21382671466683739478386787944,
1.26616234240334543615893952237, 2.45103249400615788524297078090, 3.94427902872563867956706340133, 4.37376693320301116331823968349, 6.46059914035452238497347661740, 6.69314681958515109951721339927, 7.963798780124278500615100426296, 8.778839235332680616861241843520, 9.511612604772405477019655679410, 10.18323751852492889291849094154