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
−8.218669316908950529251621970975, −7.51244722779132455488311938850, −6.64725579999132615347491504221, −6.10270164826179713246749413678, −5.35323659869098991917157244386, −4.52020567753394969491021155998, −3.82288345560584941836361164037, −2.83445810712559253034214697924, −2.10422136582708275422063904289, −0.920772175312508306128775001377,
0.46438957403768265157818910288, 1.72636298507664953735269052166, 2.49197133748881539672909304268, 3.68640903147524051846045871005, 4.16033409312504071078889926326, 4.99241485170495571128446592611, 5.86906162130743002115587692141, 6.57740782418473537831566035843, 7.12350791899728955739604573591, 7.934720996971580467658431262459