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
−10.27891480669313857189509453094, −9.571398246816521954827227754632, −8.825962325139006811078003293437, −7.80981246514330455237756856296, −7.30182771542825774183885464844, −6.42995315506966725730359593491, −5.02051227060659874778041918988, −3.91561664027990543104568426211, −2.38688589006426069518420711434, −1.30014923001792678832366058933,
1.61851662295581962446346304043, 2.66966866235780628577088116492, 3.69790458023625088044437922365, 5.05045145378014216180767632618, 6.17375660951023261013030587489, 7.41741265373063801506676556047, 8.505594209266699330750514507495, 8.722878792656982787013363491523, 9.620813940695247446383177987934, 10.54532447371038533739968261327