L(s) = 1 | − i·7-s + 11-s + 2i·13-s − 6·19-s + i·23-s + 29-s − 2·31-s − 7i·37-s + 8·41-s + i·43-s + 2i·47-s − 49-s − 14i·53-s + 10·59-s − 3i·67-s + ⋯ |
L(s) = 1 | − 0.377i·7-s + 0.301·11-s + 0.554i·13-s − 1.37·19-s + 0.208i·23-s + 0.185·29-s − 0.359·31-s − 1.15i·37-s + 1.24·41-s + 0.152i·43-s + 0.291i·47-s − 0.142·49-s − 1.92i·53-s + 1.30·59-s − 0.366i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.563705878\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.563705878\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 + 14iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 3iT - 67T^{2} \) |
| 71 | \( 1 - 9T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−7.934720996971580467658431262459, −7.12350791899728955739604573591, −6.57740782418473537831566035843, −5.86906162130743002115587692141, −4.99241485170495571128446592611, −4.16033409312504071078889926326, −3.68640903147524051846045871005, −2.49197133748881539672909304268, −1.72636298507664953735269052166, −0.46438957403768265157818910288,
0.920772175312508306128775001377, 2.10422136582708275422063904289, 2.83445810712559253034214697924, 3.82288345560584941836361164037, 4.52020567753394969491021155998, 5.35323659869098991917157244386, 6.10270164826179713246749413678, 6.64725579999132615347491504221, 7.51244722779132455488311938850, 8.218669316908950529251621970975