L(s) = 1 | − i·2-s + 3-s − 4-s + i·5-s − i·6-s − 5.12i·7-s + i·8-s + 9-s + 10-s − 3.12i·11-s − 12-s + (−0.561 + 3.56i)13-s − 5.12·14-s + i·15-s + 16-s + 2·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s − 1.93i·7-s + 0.353i·8-s + 0.333·9-s + 0.316·10-s − 0.941i·11-s − 0.288·12-s + (−0.155 + 0.987i)13-s − 1.36·14-s + 0.258i·15-s + 0.250·16-s + 0.485·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.985985 - 1.15362i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.985985 - 1.15362i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (0.561 - 3.56i)T \) |
good | 7 | \( 1 + 5.12iT - 7T^{2} \) |
| 11 | \( 1 + 3.12iT - 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 - 3.12T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 5.12iT - 31T^{2} \) |
| 37 | \( 1 - 3.12iT - 37T^{2} \) |
| 41 | \( 1 - 9.12iT - 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 10.2iT - 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 7.12iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 13.1iT - 67T^{2} \) |
| 71 | \( 1 + 6.24iT - 71T^{2} \) |
| 73 | \( 1 + 4.87iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 10.2iT - 83T^{2} \) |
| 89 | \( 1 + 5.12iT - 89T^{2} \) |
| 97 | \( 1 + 4.87iT - 97T^{2} \) |
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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
−11.07068910381417814294711411857, −10.21418346651246050474208042421, −9.469400185539040864350102951179, −8.358626386927667138613501825776, −7.34746814183212372238007293265, −6.56608982688036452895534213985, −4.71844180286973510543996675000, −3.82735430666502852902875426281, −2.83995846023222531482386603290, −1.04359650279217571606764863877,
2.05518018277053265444179179702, 3.45992713194320431476358006364, 5.10594149435196506632106662911, 5.59683933014197947420380552559, 6.91726724913634033252255118654, 8.122968279003970163132267309979, 8.589492276392033399101053931332, 9.505722884839012752060046433689, 10.28115609177778253235929906260, 12.08565703009952546539545842944