| L(s) = 1 | + 1.73i·3-s − 4.27·5-s − 2.99·9-s − 3.10i·11-s + 2.72·13-s − 7.40i·15-s + 5.09·17-s + 25.7i·19-s + 12.1i·23-s − 6.72·25-s − 5.19i·27-s + 41.0·29-s − 0.172i·31-s + 5.37·33-s − 16.9·37-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.854·5-s − 0.333·9-s − 0.282i·11-s + 0.209·13-s − 0.493i·15-s + 0.299·17-s + 1.35i·19-s + 0.529i·23-s − 0.269·25-s − 0.192i·27-s + 1.41·29-s − 0.00556i·31-s + 0.162·33-s − 0.457·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1874909157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1874909157\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 4.27T + 25T^{2} \) |
| 11 | \( 1 + 3.10iT - 121T^{2} \) |
| 13 | \( 1 - 2.72T + 169T^{2} \) |
| 17 | \( 1 - 5.09T + 289T^{2} \) |
| 19 | \( 1 - 25.7iT - 361T^{2} \) |
| 23 | \( 1 - 12.1iT - 529T^{2} \) |
| 29 | \( 1 - 41.0T + 841T^{2} \) |
| 31 | \( 1 + 0.172iT - 961T^{2} \) |
| 37 | \( 1 + 16.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 36.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 53.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 24.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 103.T + 2.80e3T^{2} \) |
| 59 | \( 1 - 29.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 32.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 17.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 76.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 99.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + 87.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 151. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 68.5T + 7.92e3T^{2} \) |
| 97 | \( 1 + 104.T + 9.40e3T^{2} \) |
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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
−9.306780359201316668667574750486, −8.260416329533054357005340169699, −8.010171617327143308198367883397, −6.99688149595600109886309487057, −6.02999106809509438965577673681, −5.31256933929846170257888713187, −4.25985956180035149288596039951, −3.70826633056065748895676598387, −2.83092708694955386537606054568, −1.36152672137530960174749568608,
0.05123440773356589925177204041, 1.13907772610078694357918020074, 2.45445249340438594214879875518, 3.31030628734734508677647587005, 4.38561654736545689488406627339, 5.05376771336871339474876722617, 6.27115585722804737981216750029, 6.79802032529762870580195371943, 7.71088478073443707699732328374, 8.149861603022210792802525856184
