| L(s) = 1 | + (−2.97 − 0.386i)3-s + (2.32 − 4.03i)7-s + (8.70 + 2.30i)9-s + (−12.4 − 7.21i)11-s + (12.2 + 21.2i)13-s + 6.56i·17-s − 33.0·19-s + (−8.48 + 11.0i)21-s + (31.9 − 18.4i)23-s + (−24.9 − 10.2i)27-s + (−17.4 − 10.0i)29-s + (6.48 + 11.2i)31-s + (34.3 + 26.2i)33-s − 19.9·37-s + (−28.3 − 68.0i)39-s + ⋯ |
| L(s) = 1 | + (−0.991 − 0.128i)3-s + (0.332 − 0.575i)7-s + (0.966 + 0.255i)9-s + (−1.13 − 0.655i)11-s + (0.945 + 1.63i)13-s + 0.386i·17-s − 1.74·19-s + (−0.403 + 0.528i)21-s + (1.38 − 0.802i)23-s + (−0.925 − 0.378i)27-s + (−0.602 − 0.348i)29-s + (0.209 + 0.362i)31-s + (1.04 + 0.796i)33-s − 0.538·37-s + (−0.726 − 1.74i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.103500600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.103500600\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.97 + 0.386i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-2.32 + 4.03i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (12.4 + 7.21i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-12.2 - 21.2i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 6.56iT - 289T^{2} \) |
| 19 | \( 1 + 33.0T + 361T^{2} \) |
| 23 | \( 1 + (-31.9 + 18.4i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (17.4 + 10.0i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-6.48 - 11.2i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 19.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-44.9 + 25.9i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (3.70 - 6.41i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-23.7 - 13.6i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 79.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-63.3 + 36.5i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (12.3 - 21.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-20.4 - 35.4i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 117. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 40.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + (11.1 - 19.2i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-116. - 67.3i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 4.69iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-69.4 + 120. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−10.40935210574788043785022853958, −9.074852501029416401781731628183, −8.341717471856620927315310302730, −7.24957507165478558264334561578, −6.51496867152232271883806002225, −5.74439201552258788003923710220, −4.61915937953625934374206333543, −3.97567205572835397591035153108, −2.22687027106084836139333486933, −0.914473482613376169637513319488,
0.54338522509183738491542461508, 2.10864188246366392094242981065, 3.46515885374388985256303966299, 4.82335195356222388540428206015, 5.38698195282220503788886453974, 6.17911299803149877534985967514, 7.26651708042985441407087697855, 8.084411978683793363973815824878, 9.008706949710247444694358157548, 10.11194905603170107974867725707
