| L(s) = 1 | − i·2-s − i·3-s − 4-s + (2 + i)5-s − 6-s − 2i·7-s + i·8-s − 9-s + (1 − 2i)10-s + 4·11-s + i·12-s − 6i·13-s − 2·14-s + (1 − 2i)15-s + 16-s + 6i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.894 + 0.447i)5-s − 0.408·6-s − 0.755i·7-s + 0.353i·8-s − 0.333·9-s + (0.316 − 0.632i)10-s + 1.20·11-s + 0.288i·12-s − 1.66i·13-s − 0.534·14-s + (0.258 − 0.516i)15-s + 0.250·16-s + 1.45i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.841651001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.841651001\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2 - i)T \) |
| 37 | \( 1 - iT \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 14iT - 53T^{2} \) |
| 59 | \( 1 + 14T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−9.875538461991572270039562668007, −8.712317486699673711262015630400, −8.137299122905775590988903394213, −6.89494957778772227203001286030, −6.29783850258393869721708026030, −5.35507438991081208800447699704, −4.05192271090513651256314678160, −3.13246285146005863052316712149, −1.97868340288436899675780708410, −0.904483457506386688431449941298,
1.52298635124479814406354454352, 2.92157440145278415652892382268, 4.46295378969176626891892718463, 4.81442077210364497922619153180, 6.13262135834501838138314047609, 6.39288926976370005862739861098, 7.59514694031345826863883307751, 8.917048079349350748800251563254, 9.192659590786573781069738731393, 9.575870782330406636055930869150
