| L(s) = 1 | + (−0.823 + 1.14i)2-s + (1.67 + 2.50i)3-s + (−0.642 − 1.89i)4-s + (−0.803 − 0.159i)5-s + (−4.25 − 0.139i)6-s + (−0.704 − 3.54i)7-s + (2.70 + 0.821i)8-s + (−2.31 + 5.59i)9-s + (0.845 − 0.791i)10-s + (1.82 + 1.21i)11-s + (3.66 − 4.77i)12-s + (−0.604 − 0.604i)13-s + (4.65 + 2.10i)14-s + (−0.943 − 2.27i)15-s + (−3.17 + 2.43i)16-s + (0.361 − 4.10i)17-s + ⋯ |
| L(s) = 1 | + (−0.582 + 0.812i)2-s + (0.965 + 1.44i)3-s + (−0.321 − 0.946i)4-s + (−0.359 − 0.0714i)5-s + (−1.73 − 0.0569i)6-s + (−0.266 − 1.33i)7-s + (0.956 + 0.290i)8-s + (−0.772 + 1.86i)9-s + (0.267 − 0.250i)10-s + (0.549 + 0.367i)11-s + (1.05 − 1.37i)12-s + (−0.167 − 0.167i)13-s + (1.24 + 0.563i)14-s + (−0.243 − 0.587i)15-s + (−0.793 + 0.608i)16-s + (0.0876 − 0.996i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0870 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0870 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.563401 + 0.614789i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.563401 + 0.614789i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.823 - 1.14i)T \) |
| 17 | \( 1 + (-0.361 + 4.10i)T \) |
| good | 3 | \( 1 + (-1.67 - 2.50i)T + (-1.14 + 2.77i)T^{2} \) |
| 5 | \( 1 + (0.803 + 0.159i)T + (4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (0.704 + 3.54i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-1.82 - 1.21i)T + (4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (0.604 + 0.604i)T + 13iT^{2} \) |
| 19 | \( 1 + (-2.75 + 1.14i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (1.97 - 2.96i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (0.860 - 4.32i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (-1.27 + 0.852i)T + (11.8 - 28.6i)T^{2} \) |
| 37 | \( 1 + (-2.06 + 1.37i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (2.36 - 0.471i)T + (37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (7.50 + 3.10i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (6.12 - 6.12i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.89 + 9.40i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (0.113 - 0.274i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-1.85 - 9.33i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 - 5.75T + 67T^{2} \) |
| 71 | \( 1 + (-0.779 - 1.16i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-11.4 - 2.27i)T + (67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-3.81 - 2.54i)T + (30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-1.97 - 4.77i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (8.86 - 8.86i)T - 89iT^{2} \) |
| 97 | \( 1 + (0.622 - 3.13i)T + (-89.6 - 37.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.25300619314100744508615775846, −14.25280974231038118198095321745, −13.62189384761969998089983451982, −11.22506537658911785992051927405, −9.963393127496284431803212468460, −9.550477013305019161888179612002, −8.175226901515660668238441870447, −7.12042988677021939900828207480, −4.92804031667738408766683903081, −3.75530126128982882440566602568,
1.97119063031566394745204456754, 3.36464106464965989345453014850, 6.37892116643777306076406529477, 7.88515644604239476911780856529, 8.570959541335978581079939039065, 9.620871893548245244338838067174, 11.62095618793709028543776121635, 12.22901573925483066089007690773, 13.11813654280233156870355147475, 14.20326700490516220414190459958
