L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + 5-s + (−1.82 + 1.91i)7-s + 0.999i·8-s + (0.866 + 0.5i)10-s + 3.07i·11-s + (−0.449 − 0.259i)13-s + (−2.53 + 0.752i)14-s + (−0.5 + 0.866i)16-s + (2.44 − 4.23i)17-s + (0.713 − 0.412i)19-s + (0.499 + 0.866i)20-s + (−1.53 + 2.66i)22-s + 8.94i·23-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + 0.447·5-s + (−0.688 + 0.725i)7-s + 0.353i·8-s + (0.273 + 0.158i)10-s + 0.927i·11-s + (−0.124 − 0.0720i)13-s + (−0.677 + 0.200i)14-s + (−0.125 + 0.216i)16-s + (0.592 − 1.02i)17-s + (0.163 − 0.0945i)19-s + (0.111 + 0.193i)20-s + (−0.328 + 0.568i)22-s + 1.86i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.072788533\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.072788533\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (1.82 - 1.91i)T \) |
good | 11 | \( 1 - 3.07iT - 11T^{2} \) |
| 13 | \( 1 + (0.449 + 0.259i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.44 + 4.23i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.713 + 0.412i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 8.94iT - 23T^{2} \) |
| 29 | \( 1 + (5.55 - 3.20i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.784 - 0.452i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.53 - 4.39i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.18 - 3.78i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.84 + 3.19i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.89 - 3.27i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.69 + 2.71i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.29 - 9.16i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (8.24 + 4.76i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.95 + 6.84i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.6iT - 71T^{2} \) |
| 73 | \( 1 + (-5.78 - 3.33i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.73 - 2.99i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.32 - 5.75i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.771 + 1.33i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.93 + 1.11i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.645543143157490188874221082562, −8.793374039839255450250810575999, −7.62480812619653016827144936594, −7.12881802108637950198535462825, −6.20705567892389521459876976307, −5.42254088224029425711669283330, −4.88797796333133466545133750766, −3.57584623585013897311300237526, −2.83540875727502359922901675103, −1.70554793868326885833195304002,
0.58869540011745525403336160661, 1.97055221072718309600698427732, 3.13192899099523560364972767730, 3.83003882010990382015994422692, 4.75183990827032289515959862214, 5.93314768366449363111514111897, 6.22409375634306167830239825239, 7.23162202173897837923905948678, 8.164601818454020815771647136766, 9.078529874936412787822666240833