L(s) = 1 | − 2.18i·2-s + i·3-s − 2.79·4-s + (2 + i)5-s + 2.18·6-s + 4.37i·7-s + 1.73i·8-s − 9-s + (2.18 − 4.37i)10-s − 2.79i·12-s + 4.37i·13-s + 9.58·14-s + (−1 + 2i)15-s − 1.79·16-s + 3.55i·17-s + 2.18i·18-s + ⋯ |
L(s) = 1 | − 1.54i·2-s + 0.577i·3-s − 1.39·4-s + (0.894 + 0.447i)5-s + 0.893·6-s + 1.65i·7-s + 0.612i·8-s − 0.333·9-s + (0.692 − 1.38i)10-s − 0.805i·12-s + 1.21i·13-s + 2.56·14-s + (−0.258 + 0.516i)15-s − 0.447·16-s + 0.863i·17-s + 0.515i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.189164343\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.189164343\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2 - i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.18iT - 2T^{2} \) |
| 7 | \( 1 - 4.37iT - 7T^{2} \) |
| 13 | \( 1 - 4.37iT - 13T^{2} \) |
| 17 | \( 1 - 3.55iT - 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 + 8.58iT - 23T^{2} \) |
| 29 | \( 1 + 0.913T + 29T^{2} \) |
| 31 | \( 1 + 6.58T + 31T^{2} \) |
| 37 | \( 1 - 0.417iT - 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 0.913iT - 43T^{2} \) |
| 47 | \( 1 + 2.58iT - 47T^{2} \) |
| 53 | \( 1 + 5iT - 53T^{2} \) |
| 59 | \( 1 + 5.58T + 59T^{2} \) |
| 61 | \( 1 + 8.66T + 61T^{2} \) |
| 67 | \( 1 - 12.7iT - 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 - 5.10iT - 73T^{2} \) |
| 79 | \( 1 - 2.64T + 79T^{2} \) |
| 83 | \( 1 + 1.63iT - 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 - 5.58iT - 97T^{2} \) |
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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
−9.544288496925272077136204937515, −8.848454605347608471035954334559, −8.581668423962115981432932261462, −6.69098319090565357448893020322, −6.12295589574099612816892299568, −5.10868456080130736300456001525, −4.24432182623221976325615265098, −3.23786805469068201103229608061, −2.18206725836976204408133016842, −1.98338407681262515928577261901,
0.40740572924664099062540399203, 1.74070479175142017632947858453, 3.35810045947812111563884862118, 4.59140083342498773021894369266, 5.32267746046364799032502480822, 6.06722142226701153063678902660, 6.80369164717836451215404253556, 7.55574417094516416452910964927, 7.893899891280124989406451707867, 8.933690296368970015738607200053