L(s) = 1 | + i·2-s + 4-s + 3i·8-s + 4·11-s + 2i·13-s − 16-s − 2i·17-s − 4·19-s + 4i·22-s − 2·26-s − 2·29-s + 5i·32-s + 2·34-s − 10i·37-s − 4i·38-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.5·4-s + 1.06i·8-s + 1.20·11-s + 0.554i·13-s − 0.250·16-s − 0.485i·17-s − 0.917·19-s + 0.852i·22-s − 0.392·26-s − 0.371·29-s + 0.883i·32-s + 0.342·34-s − 1.64i·37-s − 0.648i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23049 + 0.760489i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23049 + 0.760489i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - iT - 2T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−12.21514376540988951805356976983, −11.54674679890228818526535859466, −10.58914435162505058368393537594, −9.266239846421013409544216580952, −8.402286387150332255731445415079, −7.12591498439620466491046998578, −6.52726496378297592542480322022, −5.35071924997917086897145940885, −3.87774102242517822582424957761, −2.05931905866322724216651857628,
1.54453184988008222487892951174, 3.10508595710742423327986299334, 4.29639140720445394036584687769, 6.05518818399871603277342878313, 6.88565892634997875356151165205, 8.190105935508340286716024281852, 9.371726320094956024137926268380, 10.33956712862129372655630703634, 11.14058506421717825780592426261, 12.03101620907974968982771505105