L(s) = 1 | + (0.866 − 0.5i)3-s + (−0.866 + 2.5i)7-s + (−1 + 1.73i)9-s + (−1 − 1.73i)11-s + (−3.46 + 2i)17-s + (−1 + 1.73i)19-s + (0.500 + 2.59i)21-s + (−0.866 − 0.5i)23-s + 5i·27-s − 9·29-s + (−2 − 3.46i)31-s + (−1.73 − 0.999i)33-s + (3.46 + 2i)37-s + 41-s + 9i·43-s + ⋯ |
L(s) = 1 | + (0.499 − 0.288i)3-s + (−0.327 + 0.944i)7-s + (−0.333 + 0.577i)9-s + (−0.301 − 0.522i)11-s + (−0.840 + 0.485i)17-s + (−0.229 + 0.397i)19-s + (0.109 + 0.566i)21-s + (−0.180 − 0.104i)23-s + 0.962i·27-s − 1.67·29-s + (−0.359 − 0.622i)31-s + (−0.301 − 0.174i)33-s + (0.569 + 0.328i)37-s + 0.156·41-s + 1.37i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8438836370\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8438836370\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.866 - 2.5i)T \) |
good | 3 | \( 1 + (-0.866 + 0.5i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + (3.46 - 2i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.46 - 2i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - T + 41T^{2} \) |
| 43 | \( 1 - 9iT - 43T^{2} \) |
| 47 | \( 1 + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (8.66 - 5i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.5 - 7.79i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.33 - 2.5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 + (-10.3 + 6i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7 + 12.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 11iT - 83T^{2} \) |
| 89 | \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 18iT - 97T^{2} \) |
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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
−9.553437156040006702085224619451, −9.073084704969979318751940674833, −8.137961076354853889560259150878, −7.76114669496060172506307633190, −6.45192933827419956587209055068, −5.83179161131033372613572089833, −4.91319735809698487200322051045, −3.66271441719513265454371565313, −2.66158356379609903894073574869, −1.87203923973009453159345560663,
0.29255729232755309651972439819, 2.05511623152055733804699600409, 3.23922381633568738896739154300, 4.00272399517110502234880523777, 4.88760439311596312604560381989, 6.05563665250773354964402971804, 6.96411468009677848550885179092, 7.56879288114762012861960849665, 8.601232162727144634415163833368, 9.358320604388130446600904094603