L(s) = 1 | + (0.5 − 0.866i)3-s + 4·7-s + (1 + 1.73i)9-s − 3·11-s + (−1 − 1.73i)13-s + (3 − 5.19i)17-s + (3.5 + 2.59i)19-s + (2 − 3.46i)21-s + (−3 − 5.19i)23-s + (2.5 + 4.33i)25-s + 5·27-s − 2·31-s + (−1.5 + 2.59i)33-s − 10·37-s − 1.99·39-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + 1.51·7-s + (0.333 + 0.577i)9-s − 0.904·11-s + (−0.277 − 0.480i)13-s + (0.727 − 1.26i)17-s + (0.802 + 0.596i)19-s + (0.436 − 0.755i)21-s + (−0.625 − 1.08i)23-s + (0.5 + 0.866i)25-s + 0.962·27-s − 0.359·31-s + (−0.261 + 0.452i)33-s − 1.64·37-s − 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.56458 - 0.338148i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56458 - 0.338148i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-3.5 - 2.59i)T \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.5 - 14.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.70267975460397978759753019787, −10.73359721735879179389445418935, −9.932640092240573728227139780063, −8.464498472816492501032971920398, −7.81492517044600848278144870102, −7.18390465856659408311806978343, −5.37100781141545302116005837049, −4.80639296637173689723380906051, −2.90543307958866180738477720057, −1.55470481856806376749536800444,
1.74890550306580313603717464941, 3.49310160595025062941920708005, 4.66394074691694157525008378116, 5.56460508320998101200922413237, 7.12523186046341795950573104520, 8.064793730236352487603491001056, 8.879975529068642090755983407217, 10.03159301912368317158478020777, 10.71847297110403078461802098912, 11.79046945920849194084307987192