L(s) = 1 | + (0.5 − 0.133i)3-s + (−0.133 − 0.232i)7-s + (−2.36 + 1.36i)9-s + (4.59 − 1.23i)11-s + (−2 − 3i)13-s + (0.767 − 2.86i)17-s + (0.866 − 3.23i)19-s + (−0.0980 − 0.0980i)21-s + (0.0358 + 0.133i)23-s + (−2.09 + 2.09i)27-s + (1.03 + 0.598i)29-s + (2.26 − 2.26i)31-s + (2.13 − 1.23i)33-s + (1.86 − 3.23i)37-s + (−1.40 − 1.23i)39-s + ⋯ |
L(s) = 1 | + (0.288 − 0.0773i)3-s + (−0.0506 − 0.0877i)7-s + (−0.788 + 0.455i)9-s + (1.38 − 0.371i)11-s + (−0.554 − 0.832i)13-s + (0.186 − 0.695i)17-s + (0.198 − 0.741i)19-s + (−0.0214 − 0.0214i)21-s + (0.00748 + 0.0279i)23-s + (−0.403 + 0.403i)27-s + (0.192 + 0.111i)29-s + (0.407 − 0.407i)31-s + (0.371 − 0.214i)33-s + (0.306 − 0.531i)37-s + (−0.224 − 0.197i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.689783620\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.689783620\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (2 + 3i)T \) |
good | 3 | \( 1 + (-0.5 + 0.133i)T + (2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (0.133 + 0.232i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.59 + 1.23i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-0.767 + 2.86i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 3.23i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.0358 - 0.133i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-1.03 - 0.598i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.26 + 2.26i)T - 31iT^{2} \) |
| 37 | \( 1 + (-1.86 + 3.23i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.86 + 6.96i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-9.96 - 2.66i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 - 7.46T + 47T^{2} \) |
| 53 | \( 1 + (-8.46 - 8.46i)T + 53iT^{2} \) |
| 59 | \( 1 + (13.7 + 3.69i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (10.7 + 6.23i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.86 - 0.767i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + 0.928iT - 73T^{2} \) |
| 79 | \( 1 + 11.4iT - 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 + (-0.794 - 2.96i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (2.13 - 1.23i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.166633513536393044734372748200, −8.997683413325022241228626319057, −7.81132294469603213865517673639, −7.23912781104769724512233302606, −6.14244634781796097174162653011, −5.38958447625028017282321185429, −4.33522995104791983786303262190, −3.23204850639529426093298874515, −2.38060152960421022031031006421, −0.74334641785042497390203144275,
1.35921956322910521343344334389, 2.62723058252487468202790168619, 3.76540583735124761867465640936, 4.45297523352078637322337427233, 5.78563898306963835035772386326, 6.42443340858370305290149878451, 7.30740319050932493114862058848, 8.307334501040540851354438882598, 9.059063993960680621219671319818, 9.574842033975047059864423181489