L(s) = 1 | + (0.707 − 0.707i)2-s + (−2.99 − 0.152i)3-s + 3i·4-s + (4.24 − 2.63i)5-s + (−2.22 + 2.01i)6-s + (4.33 + 5.49i)7-s + (4.94 + 4.94i)8-s + (8.95 + 0.915i)9-s + (1.13 − 4.86i)10-s − 13.9i·11-s + (0.458 − 8.98i)12-s + (14.6 + 14.6i)13-s + (6.95 + 0.822i)14-s + (−13.1 + 7.25i)15-s − 4.99·16-s + (4.86 + 4.86i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.353i)2-s + (−0.998 − 0.0509i)3-s + 0.750i·4-s + (0.849 − 0.527i)5-s + (−0.371 + 0.335i)6-s + (0.619 + 0.785i)7-s + (0.618 + 0.618i)8-s + (0.994 + 0.101i)9-s + (0.113 − 0.486i)10-s − 1.26i·11-s + (0.0381 − 0.749i)12-s + (1.12 + 1.12i)13-s + (0.496 + 0.0587i)14-s + (−0.875 + 0.483i)15-s − 0.312·16-s + (0.286 + 0.286i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.154i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.46794 + 0.114140i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.46794 + 0.114140i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.99 + 0.152i)T \) |
| 5 | \( 1 + (-4.24 + 2.63i)T \) |
| 7 | \( 1 + (-4.33 - 5.49i)T \) |
good | 2 | \( 1 + (-0.707 + 0.707i)T - 4iT^{2} \) |
| 11 | \( 1 + 13.9iT - 121T^{2} \) |
| 13 | \( 1 + (-14.6 - 14.6i)T + 169iT^{2} \) |
| 17 | \( 1 + (-4.86 - 4.86i)T + 289iT^{2} \) |
| 19 | \( 1 + 21.7T + 361T^{2} \) |
| 23 | \( 1 + (-1.77 - 1.77i)T + 529iT^{2} \) |
| 29 | \( 1 + 28.0T + 841T^{2} \) |
| 31 | \( 1 + 17.2iT - 961T^{2} \) |
| 37 | \( 1 + (6.50 + 6.50i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 26.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-33.1 + 33.1i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (18.5 + 18.5i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (48.3 + 48.3i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 29.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 21.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (32.4 + 32.4i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 16.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-57.3 - 57.3i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 75.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (51.9 - 51.9i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 174. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (16.6 - 16.6i)T - 9.40e3iT^{2} \) |
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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
−13.26425375032032050587051222743, −12.51124498565361988014020387675, −11.43124183417459338917921147700, −10.91211999459489780532539125482, −9.124050570504096771648159299501, −8.230662175457822890581912993063, −6.37300273117808278216897385922, −5.43398939493534927974766409194, −4.08239974258686348291179754543, −1.86254738618732232498433600067,
1.43908391472416274587895741440, 4.37765822038827050117005309851, 5.50821008188516714987204097604, 6.45287944292565890891807658437, 7.46588160452352546623518354175, 9.624233235198907204407054992870, 10.59455604138580791259777836246, 10.92650325008719483706242022466, 12.71371653554167503409595929396, 13.50096837638738149849832999351