| L(s) = 1 | + (−3.67 + 2.12i)5-s + (3.5 + 6.06i)7-s + (−18.3 − 10.6i)11-s + 23·13-s + (14.6 + 8.48i)17-s + (−0.5 − 0.866i)19-s + (14.6 − 8.48i)23-s + (−3.5 + 6.06i)25-s + 33.9i·29-s + (−24.5 + 42.4i)31-s + (−25.7 − 14.8i)35-s + (−8.5 − 14.7i)37-s − 21.2i·41-s − 47·43-s + (−33.0 + 19.0i)47-s + ⋯ |
| L(s) = 1 | + (−0.734 + 0.424i)5-s + (0.5 + 0.866i)7-s + (−1.67 − 0.964i)11-s + 1.76·13-s + (0.864 + 0.499i)17-s + (−0.0263 − 0.0455i)19-s + (0.638 − 0.368i)23-s + (−0.140 + 0.242i)25-s + 1.17i·29-s + (−0.790 + 1.36i)31-s + (−0.734 − 0.424i)35-s + (−0.229 − 0.397i)37-s − 0.517i·41-s − 1.09·43-s + (−0.703 + 0.406i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9118730687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9118730687\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-3.5 - 6.06i)T \) |
| good | 5 | \( 1 + (3.67 - 2.12i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (18.3 + 10.6i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 23T + 169T^{2} \) |
| 17 | \( 1 + (-14.6 - 8.48i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-14.6 + 8.48i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 33.9iT - 841T^{2} \) |
| 31 | \( 1 + (24.5 - 42.4i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (8.5 + 14.7i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 21.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 47T + 1.84e3T^{2} \) |
| 47 | \( 1 + (33.0 - 19.0i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (73.4 + 42.4i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (44.0 + 25.4i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-20 - 34.6i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-11.5 + 19.9i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 63.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (8.5 - 14.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (39.5 + 68.4i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 106. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (117. - 67.8i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 40T + 9.40e3T^{2} \) |
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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
−10.40689860231359354234489690331, −8.968911794265952926420939478111, −8.348585537019708272827160657883, −7.86452157991361868605119793163, −6.71112332183913033110448324858, −5.64926225531785140505073396526, −5.12032359666615993612800738529, −3.54094889400266365484414886294, −3.03942414869655281741441763602, −1.45550120393563986383252144314,
0.29709721381251255088277996491, 1.60700679005466256741153799480, 3.16365209515022091305171681963, 4.18232806504292397265300267415, 4.91909357842074048255224775997, 5.91782199641265645199256863764, 7.18313126332856973411640039613, 7.943304636325081029848042529187, 8.221810045555883752746907696458, 9.566752652872818592122640849935
