| L(s) = 1 | − 3.10i·3-s − 2.52i·5-s − 7-s − 6.62·9-s − 3.62i·11-s − 4.72i·13-s − 7.83·15-s + 4.20·17-s − 7.10i·19-s + 3.10i·21-s − 0.578·23-s − 1.37·25-s + 11.2i·27-s + 8.20i·29-s + 5.04·31-s + ⋯ |
| L(s) = 1 | − 1.79i·3-s − 1.12i·5-s − 0.377·7-s − 2.20·9-s − 1.09i·11-s − 1.31i·13-s − 2.02·15-s + 1.01·17-s − 1.62i·19-s + 0.677i·21-s − 0.120·23-s − 0.274·25-s + 2.16i·27-s + 1.52i·29-s + 0.906·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.449759970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.449759970\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 3.10iT - 3T^{2} \) |
| 5 | \( 1 + 2.52iT - 5T^{2} \) |
| 11 | \( 1 + 3.62iT - 11T^{2} \) |
| 13 | \( 1 + 4.72iT - 13T^{2} \) |
| 17 | \( 1 - 4.20T + 17T^{2} \) |
| 19 | \( 1 + 7.10iT - 19T^{2} \) |
| 23 | \( 1 + 0.578T + 23T^{2} \) |
| 29 | \( 1 - 8.20iT - 29T^{2} \) |
| 31 | \( 1 - 5.04T + 31T^{2} \) |
| 37 | \( 1 - 3.04iT - 37T^{2} \) |
| 41 | \( 1 - 0.205T + 41T^{2} \) |
| 43 | \( 1 + 4.78iT - 43T^{2} \) |
| 47 | \( 1 - 6.20T + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 12.5iT - 59T^{2} \) |
| 61 | \( 1 - 10.5iT - 61T^{2} \) |
| 67 | \( 1 - 6.57iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 9.25T + 73T^{2} \) |
| 79 | \( 1 - 5.15T + 79T^{2} \) |
| 83 | \( 1 - 5.94iT - 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 9.36T + 97T^{2} \) |
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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
−8.556903943000777115038639054886, −8.062360831544911399372915146731, −7.24618471332696302069481860385, −6.46146117844499068677614306882, −5.59666400471890449980018815066, −5.07715029281113409831133673537, −3.35076308891791777831293569350, −2.60360941982545705466697195534, −1.07908825927318766029820885002, −0.64636303982010474298800242245,
2.19211821425815739100852421239, 3.24305800829473832057162253256, 3.98545578262573783185614060893, 4.59476818377590432635730011699, 5.74972753648972529159623219089, 6.39992977233944600317743299366, 7.43014501858267193935020036169, 8.304712562145747115670821745897, 9.526197606365709349018940230141, 9.710946830472330416106909765323
