L(s) = 1 | + (−1 + 1.73i)2-s + (1.5 − 2.59i)3-s + (−1.99 − 3.46i)4-s + (4.22 − 7.31i)5-s + (3 + 5.19i)6-s − 9.44·7-s + 7.99·8-s + (−4.5 − 7.79i)9-s + (8.44 + 14.6i)10-s − 47.4·11-s − 12·12-s + (−33.8 − 58.5i)13-s + (9.44 − 16.3i)14-s + (−12.6 − 21.9i)15-s + (−8 + 13.8i)16-s + (38.8 − 67.2i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.377 − 0.653i)5-s + (0.204 + 0.353i)6-s − 0.509·7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.266 + 0.462i)10-s − 1.30·11-s − 0.288·12-s + (−0.721 − 1.25i)13-s + (0.180 − 0.312i)14-s + (−0.217 − 0.377i)15-s + (−0.125 + 0.216i)16-s + (0.553 − 0.958i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.642043 - 0.730723i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.642043 - 0.730723i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - 1.73i)T \) |
| 3 | \( 1 + (-1.5 + 2.59i)T \) |
| 19 | \( 1 + (-53.6 - 63.0i)T \) |
good | 5 | \( 1 + (-4.22 + 7.31i)T + (-62.5 - 108. i)T^{2} \) |
| 7 | \( 1 + 9.44T + 343T^{2} \) |
| 11 | \( 1 + 47.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + (33.8 + 58.5i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-38.8 + 67.2i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 23 | \( 1 + (85.4 + 148. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (120. + 207. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 279.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 20.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + (35.9 - 62.1i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (73.6 - 127. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-103. - 179. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-130. - 226. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-10.9 + 19.0i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-326. - 565. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-206. - 358. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-78.9 + 136. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-551. + 954. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-208. + 361. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.43e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (644. + 1.11e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (69.3 - 120. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.00149578233915804130938340510, −12.10666243393215125309342141034, −10.26668448169547944786057924698, −9.618007897858135685122099347912, −8.183872927012504727543129812855, −7.58235215083468410828432994107, −6.02897160987068082314794551288, −5.03318559707406621169362110875, −2.72742185328317744655246672136, −0.54822592171549979973705404020,
2.27718825712673483908890136924, 3.51707190987950777685456118786, 5.18329193318385472835001984799, 6.87305315639958888978941993333, 8.139717975497518009442887125399, 9.522234464913744179349887278011, 10.09597362438677454078703116106, 11.06952486098693986757438078660, 12.24241798143582346239468419122, 13.43048249988944526285127844220