| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + 2i·7-s + i·8-s − 9-s − 4·11-s − i·12-s + 4i·13-s + 2·14-s + 16-s − 6i·17-s + i·18-s − 2·21-s + 4i·22-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.755i·7-s + 0.353i·8-s − 0.333·9-s − 1.20·11-s − 0.288i·12-s + 1.10i·13-s + 0.534·14-s + 0.250·16-s − 1.45i·17-s + 0.235i·18-s − 0.436·21-s + 0.852i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6168652856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6168652856\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 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
−8.297165895763626958485109469793, −7.47352824506083172324628333203, −6.58587735780845912291159579660, −5.55467420874670697731438341825, −5.02738446201542419047837315429, −4.38224063713960203548380550752, −3.30524515141405925393753098321, −2.64808119993229868427874160573, −1.83801307474360219688618441267, −0.19610682539778125848197742556,
0.947175632973821499757003503611, 2.19700844669202598010403222586, 3.29364746666394977663632403858, 4.06846148656211302832271321990, 5.11178401736129917960002423638, 5.71208491667958409460348945878, 6.35945644670303305561379505766, 7.32336264006403930352848528215, 7.68262004501480624743289069014, 8.283732633385431767123709470334
