| L(s) = 1 | + 5.19·3-s − 39.7i·5-s + 46.0i·7-s + 27·9-s − 181.·11-s + 183. i·13-s − 206. i·15-s + 427.·17-s − 668.·19-s + 239. i·21-s + 882. i·23-s − 954.·25-s + 140.·27-s + 807. i·29-s − 391. i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.58i·5-s + 0.939i·7-s + 0.333·9-s − 1.49·11-s + 1.08i·13-s − 0.917i·15-s + 1.48·17-s − 1.85·19-s + 0.542i·21-s + 1.66i·23-s − 1.52·25-s + 0.192·27-s + 0.959i·29-s − 0.407i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.327092782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.327092782\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 5.19T \) |
| good | 5 | \( 1 + 39.7iT - 625T^{2} \) |
| 7 | \( 1 - 46.0iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 181.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 183. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 427.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 668.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 882. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 807. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 391. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 466. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.15e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 509.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.05e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 753. iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 1.30e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 801. iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 505.T + 2.01e7T^{2} \) |
| 71 | \( 1 + 2.17e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 2.29e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.07e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 2.97e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 6.49e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 8.18e3T + 8.85e7T^{2} \) |
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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
−10.94191174553275789266331671730, −9.679983711615281883469513549883, −9.050196133608354426373648107177, −8.300539128192842198391296233671, −7.55544970752286158671923888400, −5.84448701740718911128584637980, −5.13146819583543405811115973431, −4.06129343924423344890375528613, −2.51713201098634714734369018601, −1.40402525523466349098537537724,
0.33753040508767507228070569857, 2.42271228042856419361943939839, 3.10232688147148742536401163201, 4.27581227997440029755409440735, 5.80903065844280944558572229380, 6.86559717012489539656356613364, 7.73389620397500756213538838175, 8.276934706007550619356383334626, 10.07813817157796162959739427557, 10.42577232346776681404522263870
