L(s) = 1 | + (−0.796 + 0.796i)3-s + 1.53i·5-s + (0.707 − 0.707i)7-s + 1.73i·9-s + (2.10 − 2.10i)11-s + (3.47 − 3.47i)13-s + (−1.22 − 1.22i)15-s + (1.03 + 1.03i)17-s + (2.78 + 2.78i)19-s + 1.12i·21-s − 4.78·23-s + 2.62·25-s + (−3.76 − 3.76i)27-s + (1.87 − 1.87i)29-s + 1.64·31-s + ⋯ |
L(s) = 1 | + (−0.460 + 0.460i)3-s + 0.688i·5-s + (0.267 − 0.267i)7-s + 0.576i·9-s + (0.635 − 0.635i)11-s + (0.964 − 0.964i)13-s + (−0.316 − 0.316i)15-s + (0.250 + 0.250i)17-s + (0.638 + 0.638i)19-s + 0.245i·21-s − 0.998·23-s + 0.525·25-s + (−0.725 − 0.725i)27-s + (0.347 − 0.347i)29-s + 0.295·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.496 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.542508321\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.542508321\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
| 41 | \( 1 + (5.27 - 3.63i)T \) |
good | 3 | \( 1 + (0.796 - 0.796i)T - 3iT^{2} \) |
| 5 | \( 1 - 1.53iT - 5T^{2} \) |
| 11 | \( 1 + (-2.10 + 2.10i)T - 11iT^{2} \) |
| 13 | \( 1 + (-3.47 + 3.47i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.03 - 1.03i)T + 17iT^{2} \) |
| 19 | \( 1 + (-2.78 - 2.78i)T + 19iT^{2} \) |
| 23 | \( 1 + 4.78T + 23T^{2} \) |
| 29 | \( 1 + (-1.87 + 1.87i)T - 29iT^{2} \) |
| 31 | \( 1 - 1.64T + 31T^{2} \) |
| 37 | \( 1 - 7.76T + 37T^{2} \) |
| 43 | \( 1 - 7.65iT - 43T^{2} \) |
| 47 | \( 1 + (-6.34 - 6.34i)T + 47iT^{2} \) |
| 53 | \( 1 + (3.97 - 3.97i)T - 53iT^{2} \) |
| 59 | \( 1 + 4.91T + 59T^{2} \) |
| 61 | \( 1 - 14.5iT - 61T^{2} \) |
| 67 | \( 1 + (11.3 + 11.3i)T + 67iT^{2} \) |
| 71 | \( 1 + (-6.65 + 6.65i)T - 71iT^{2} \) |
| 73 | \( 1 - 12.2iT - 73T^{2} \) |
| 79 | \( 1 + (-2.18 + 2.18i)T - 79iT^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + (-11.2 + 11.2i)T - 89iT^{2} \) |
| 97 | \( 1 + (-12.3 - 12.3i)T + 97iT^{2} \) |
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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
−10.28023546258442135521477177366, −9.230582066323432119291135928242, −8.057772058652577406493575680824, −7.71795288039483311647125164196, −6.23801531296872444265627552945, −5.93687482017770804373080092655, −4.75038313852995140820498394211, −3.78808367667669906040120212885, −2.85737517136193751794053137044, −1.20808901918651639494344306610,
0.895294003973373486423654544764, 1.91938429446615480135859620482, 3.57227132695960166230536419436, 4.51717390683476195615332653169, 5.44696135057167139241418741729, 6.41449479039926866917222888896, 6.97239719524545817020822536558, 8.057939917975784243919326108732, 9.056694414803823983908875865451, 9.338282946765510271560722242386