L(s) = 1 | − 1.41i·2-s − 1.73·3-s − 2.00·4-s + 2.44i·6-s + (−6.63 + 2.24i)7-s + 2.82i·8-s + 2.99·9-s − 10.2·11-s + 3.46·12-s + 8.95·13-s + (3.17 + 9.37i)14-s + 4.00·16-s + 30.4·17-s − 4.24i·18-s + 16.1i·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.500·4-s + 0.408i·6-s + (−0.947 + 0.320i)7-s + 0.353i·8-s + 0.333·9-s − 0.931·11-s + 0.288·12-s + 0.689·13-s + (0.226 + 0.669i)14-s + 0.250·16-s + 1.78·17-s − 0.235i·18-s + 0.849i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.704 + 0.710i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.704 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7278578918\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7278578918\) |
\(L(2)\) |
|
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 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (6.63 - 2.24i)T \) |
good | 11 | \( 1 + 10.2T + 121T^{2} \) |
| 13 | \( 1 - 8.95T + 169T^{2} \) |
| 17 | \( 1 - 30.4T + 289T^{2} \) |
| 19 | \( 1 - 16.1iT - 361T^{2} \) |
| 23 | \( 1 + 6.72iT - 529T^{2} \) |
| 29 | \( 1 + 30T + 841T^{2} \) |
| 31 | \( 1 - 50.1iT - 961T^{2} \) |
| 37 | \( 1 + 30.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 7.10iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 74.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 58.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 70.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 0.492iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 2.86iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 27.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 50.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 70.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 133.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 104.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 144. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 100.T + 9.40e3T^{2} \) |
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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
−9.740805878231564732357324558050, −8.709684556166887696377057788381, −7.84183883127947374689174663445, −6.80671142028236549399793959585, −5.67436576933544405103724081460, −5.30101551140807399791531963714, −3.77377914386260241166981363021, −3.14329567416303232623150966748, −1.72333611964971662373658727119, −0.30817409510763050177991686380,
0.980271013007225842991207609547, 2.95154787015763153053191718400, 3.94191228401396976259769361943, 5.10975496389236248816796322825, 5.85930678868484271742666587989, 6.53358830637592570700656389712, 7.55917788060345809055774472234, 8.047186231256228628900608246685, 9.420613606970550139338673893098, 9.823138493445799262116534701221