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
−10.07617738862832740868772997696, −9.635715393698191894320210122614, −8.270131603709969115419883108435, −7.56067348418495319183702337294, −6.75997949416552802853178661534, −6.02764031183764094831082305862, −4.82054785071729600722361054051, −3.61689096366464833332730285119, −2.69566107516538558950744294926, −0.27956317219564488293696167930,
1.09728548394474560499072831027, 2.76425376058821452386557076778, 3.69427099023468259397325143625, 4.87838164397002401355296349845, 5.64634373871879647477217354138, 7.07756265755608667823714266388, 7.984870815294491548235284842754, 8.910037286484597676230025521669, 9.814455903682383495500297953663, 10.18849631710124743647322952475