L(s) = 1 | + i·3-s + (−2 − i)5-s + 2·9-s − 11-s − i·13-s + (1 − 2i)15-s + 3i·17-s − 4·19-s − 2i·23-s + (3 + 4i)25-s + 5i·27-s + 29-s + 6·31-s − i·33-s + 2i·37-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (−0.894 − 0.447i)5-s + 0.666·9-s − 0.301·11-s − 0.277i·13-s + (0.258 − 0.516i)15-s + 0.727i·17-s − 0.917·19-s − 0.417i·23-s + (0.600 + 0.800i)25-s + 0.962i·27-s + 0.185·29-s + 1.07·31-s − 0.174i·33-s + 0.328i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.333883856\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.333883856\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - iT - 3T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 - 14iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 - 19iT - 97T^{2} \) |
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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
−9.222422968800396224412836216707, −8.536677345833429951867614839856, −7.87531002249553070606636250104, −7.06836063240099952703718723183, −6.12099001509499741255267768934, −5.06130092412691465277068948515, −4.30783774599762127389943430788, −3.79022341719444720541010835395, −2.54131173833178097868470506294, −1.02379750656731340639668614838,
0.60425216495940544797012983440, 2.05328371863018114736947674458, 3.06396650645317043376460911432, 4.15514975812277907812288964179, 4.78967635232577585948693001257, 6.13082086016116177721583784792, 6.78346923116214863583088022875, 7.52800915619795128245450998189, 8.004807196360362903996573534881, 8.942764804799642424343641100245