L(s) = 1 | + (0.938 − 2.84i)3-s + (4.88 − 1.05i)5-s − 6.81i·7-s + (−7.23 − 5.34i)9-s − 7.52i·11-s + 16.2i·13-s + (1.58 − 14.9i)15-s + 4.11·17-s − 7.86·19-s + (−19.4 − 6.39i)21-s + 19.5·23-s + (22.7 − 10.2i)25-s + (−22.0 + 15.6i)27-s − 55.8i·29-s − 43.4·31-s + ⋯ |
L(s) = 1 | + (0.312 − 0.949i)3-s + (0.977 − 0.210i)5-s − 0.973i·7-s + (−0.804 − 0.594i)9-s − 0.684i·11-s + 1.24i·13-s + (0.105 − 0.994i)15-s + 0.242·17-s − 0.413·19-s + (−0.924 − 0.304i)21-s + 0.848·23-s + (0.911 − 0.411i)25-s + (−0.815 + 0.578i)27-s − 1.92i·29-s − 1.40·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 + 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.27287 - 1.41559i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.27287 - 1.41559i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.938 + 2.84i)T \) |
| 5 | \( 1 + (-4.88 + 1.05i)T \) |
good | 7 | \( 1 + 6.81iT - 49T^{2} \) |
| 11 | \( 1 + 7.52iT - 121T^{2} \) |
| 13 | \( 1 - 16.2iT - 169T^{2} \) |
| 17 | \( 1 - 4.11T + 289T^{2} \) |
| 19 | \( 1 + 7.86T + 361T^{2} \) |
| 23 | \( 1 - 19.5T + 529T^{2} \) |
| 29 | \( 1 + 55.8iT - 841T^{2} \) |
| 31 | \( 1 + 43.4T + 961T^{2} \) |
| 37 | \( 1 + 31.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 51.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 51.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 61.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 82.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 97.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 4.13T + 3.72e3T^{2} \) |
| 67 | \( 1 + 63.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 40.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 78.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 51.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + 2.72T + 6.88e3T^{2} \) |
| 89 | \( 1 - 70.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 3.44iT - 9.40e3T^{2} \) |
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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
−11.68669580382662050913607033557, −10.77296557733826170712609209240, −9.532894075290828571776707305140, −8.764595046248849325071486426857, −7.55181679535619273472125215136, −6.63472719894424126920864440242, −5.72698640910203689600564491060, −4.07220006359574597880419048412, −2.41405479214756510047495721177, −1.04354724448697656679772341612,
2.22459944930654995620920580826, 3.36138716069819525911318356681, 5.14564139833729948702269461815, 5.62508024232739430740671400735, 7.16506028918208060047457370142, 8.672013500576773299173348102537, 9.202184605365891543986011010523, 10.30868024983771027229275003889, 10.79018480794268920447149459922, 12.26682438543095991587974360770