| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + (−2.40 − 4.38i)5-s + (−6.94 + 0.853i)7-s + 2.82i·8-s + (−6.20 + 3.39i)10-s − 2.88·11-s − 13.8·13-s + (1.20 + 9.82i)14-s + 4.00·16-s + 24.1·17-s − 6.53i·19-s + (4.80 + 8.76i)20-s + 4.07i·22-s + 28.8i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + (−0.480 − 0.876i)5-s + (−0.992 + 0.121i)7-s + 0.353i·8-s + (−0.620 + 0.339i)10-s − 0.261·11-s − 1.06·13-s + (0.0861 + 0.701i)14-s + 0.250·16-s + 1.42·17-s − 0.343i·19-s + (0.240 + 0.438i)20-s + 0.185i·22-s + 1.25i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6452645157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6452645157\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.40 + 4.38i)T \) |
| 7 | \( 1 + (6.94 - 0.853i)T \) |
| good | 11 | \( 1 + 2.88T + 121T^{2} \) |
| 13 | \( 1 + 13.8T + 169T^{2} \) |
| 17 | \( 1 - 24.1T + 289T^{2} \) |
| 19 | \( 1 + 6.53iT - 361T^{2} \) |
| 23 | \( 1 - 28.8iT - 529T^{2} \) |
| 29 | \( 1 - 32.9T + 841T^{2} \) |
| 31 | \( 1 - 2.43iT - 961T^{2} \) |
| 37 | \( 1 - 50.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 21.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 13.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 40.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 17.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 1.47iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 111. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 120. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 90.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 21.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 66.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 78.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + 90.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 44.1T + 9.40e3T^{2} \) |
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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
−10.18849631710124743647322952475, −9.814455903682383495500297953663, −8.910037286484597676230025521669, −7.984870815294491548235284842754, −7.07756265755608667823714266388, −5.64634373871879647477217354138, −4.87838164397002401355296349845, −3.69427099023468259397325143625, −2.76425376058821452386557076778, −1.09728548394474560499072831027,
0.27956317219564488293696167930, 2.69566107516538558950744294926, 3.61689096366464833332730285119, 4.82054785071729600722361054051, 6.02764031183764094831082305862, 6.75997949416552802853178661534, 7.56067348418495319183702337294, 8.270131603709969115419883108435, 9.635715393698191894320210122614, 10.07617738862832740868772997696
