L(s) = 1 | + (−0.866 − 0.5i)2-s + (1.64 + 0.542i)3-s + (0.499 + 0.866i)4-s + 5-s + (−1.15 − 1.29i)6-s + (1.49 + 2.18i)7-s − 0.999i·8-s + (2.41 + 1.78i)9-s + (−0.866 − 0.5i)10-s − 3.47i·11-s + (0.353 + 1.69i)12-s + (0.972 + 0.561i)13-s + (−0.205 − 2.63i)14-s + (1.64 + 0.542i)15-s + (−0.5 + 0.866i)16-s + (−0.795 + 1.37i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.949 + 0.312i)3-s + (0.249 + 0.433i)4-s + 0.447·5-s + (−0.470 − 0.527i)6-s + (0.565 + 0.824i)7-s − 0.353i·8-s + (0.804 + 0.594i)9-s + (−0.273 − 0.158i)10-s − 1.04i·11-s + (0.101 + 0.489i)12-s + (0.269 + 0.155i)13-s + (−0.0550 − 0.704i)14-s + (0.424 + 0.139i)15-s + (−0.125 + 0.216i)16-s + (−0.192 + 0.334i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.76652 + 0.236483i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.76652 + 0.236483i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.64 - 0.542i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-1.49 - 2.18i)T \) |
good | 11 | \( 1 + 3.47iT - 11T^{2} \) |
| 13 | \( 1 + (-0.972 - 0.561i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.795 - 1.37i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.478 + 0.276i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 0.00487iT - 23T^{2} \) |
| 29 | \( 1 + (-2.13 + 1.22i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (5.51 - 3.18i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.212 - 0.368i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.76 - 3.05i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.300 + 0.520i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.22 + 9.05i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.21 + 1.28i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.70 - 2.94i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.05 + 2.34i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.77 - 11.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14.9iT - 71T^{2} \) |
| 73 | \( 1 + (4.45 + 2.57i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.37 + 4.11i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.21 - 5.57i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.59 + 6.23i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-14.3 + 8.26i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.54532447371038533739968261327, −9.620813940695247446383177987934, −8.722878792656982787013363491523, −8.505594209266699330750514507495, −7.41741265373063801506676556047, −6.17375660951023261013030587489, −5.05045145378014216180767632618, −3.69790458023625088044437922365, −2.66966866235780628577088116492, −1.61851662295581962446346304043,
1.30014923001792678832366058933, 2.38688589006426069518420711434, 3.91561664027990543104568426211, 5.02051227060659874778041918988, 6.42995315506966725730359593491, 7.30182771542825774183885464844, 7.80981246514330455237756856296, 8.825962325139006811078003293437, 9.571398246816521954827227754632, 10.27891480669313857189509453094