L(s) = 1 | + (4.76 − 2.75i)2-s + (2.56 + 4.51i)3-s + (11.1 − 19.3i)4-s + (10.5 + 6.07i)5-s + (24.6 + 14.4i)6-s + (−15.9 + 9.18i)7-s − 78.7i·8-s + (−13.7 + 23.2i)9-s + 66.9·10-s + (−21.1 + 12.2i)11-s + (115. + 0.746i)12-s + (−44.8 − 13.4i)13-s + (−50.5 + 87.6i)14-s + (−0.406 + 63.1i)15-s + (−127. − 221. i)16-s + 36.2·17-s + ⋯ |
L(s) = 1 | + (1.68 − 0.973i)2-s + (0.494 + 0.869i)3-s + (1.39 − 2.41i)4-s + (0.941 + 0.543i)5-s + (1.67 + 0.984i)6-s + (−0.859 + 0.496i)7-s − 3.48i·8-s + (−0.511 + 0.859i)9-s + 2.11·10-s + (−0.579 + 0.334i)11-s + (2.78 + 0.0179i)12-s + (−0.957 − 0.287i)13-s + (−0.965 + 1.67i)14-s + (−0.00699 + 1.08i)15-s + (−1.99 − 3.45i)16-s + 0.516·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.31717 - 1.46344i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.31717 - 1.46344i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.56 - 4.51i)T \) |
| 13 | \( 1 + (44.8 + 13.4i)T \) |
good | 2 | \( 1 + (-4.76 + 2.75i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (-10.5 - 6.07i)T + (62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (15.9 - 9.18i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (21.1 - 12.2i)T + (665.5 - 1.15e3i)T^{2} \) |
| 17 | \( 1 - 36.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 139. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-4.44 + 7.69i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-107. - 185. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-67.1 - 38.7i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 94.1iT - 5.06e4T^{2} \) |
| 41 | \( 1 + (26.0 + 15.0i)T + (3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-132. - 229. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (54.3 - 31.3i)T + (5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 555.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (126. + 73.0i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-386. - 669. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-19.1 - 11.0i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 820. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 267. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-66.9 - 115. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-239. + 138. i)T + (2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 1.18e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (717. - 414. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.11486878172473978375565771675, −12.13845336840355590687601393577, −10.79906481009490389576429925274, −10.10512339132640438131573733243, −9.403128618901217700238839964611, −6.83416200479226144051185518143, −5.57901910632959235155290114554, −4.71227865970351655161624021717, −2.98488741652052060904851550373, −2.52608903187094127667705880515,
2.38581495046236775982008881219, 3.75481089698351126812406241230, 5.45326951698316330984723775224, 6.24374507523872516726845550356, 7.32069370140042033944458687799, 8.270700383084440335922116137590, 9.883104380337034981284445093566, 11.93893072742071842502359182157, 12.63477620202406027424988336449, 13.37440780750552430451574184258