L(s) = 1 | + (1 − i)2-s + (0.199 + 2.99i)3-s − 2i·4-s + (4.37 − 2.41i)5-s + (3.19 + 2.79i)6-s + (4.76 − 5.12i)7-s + (−2 − 2i)8-s + (−8.92 + 1.19i)9-s + (1.95 − 6.79i)10-s − 6.70i·11-s + (5.98 − 0.398i)12-s + (16.0 + 16.0i)13-s + (−0.359 − 9.89i)14-s + (8.11 + 12.6i)15-s − 4·16-s + (−7.21 − 7.21i)17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.5i)2-s + (0.0664 + 0.997i)3-s − 0.5i·4-s + (0.875 − 0.483i)5-s + (0.532 + 0.465i)6-s + (0.680 − 0.732i)7-s + (−0.250 − 0.250i)8-s + (−0.991 + 0.132i)9-s + (0.195 − 0.679i)10-s − 0.609i·11-s + (0.498 − 0.0332i)12-s + (1.23 + 1.23i)13-s + (−0.0256 − 0.706i)14-s + (0.540 + 0.841i)15-s − 0.250·16-s + (−0.424 − 0.424i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.35959 - 0.489953i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.35959 - 0.489953i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 + (-0.199 - 2.99i)T \) |
| 5 | \( 1 + (-4.37 + 2.41i)T \) |
| 7 | \( 1 + (-4.76 + 5.12i)T \) |
good | 11 | \( 1 + 6.70iT - 121T^{2} \) |
| 13 | \( 1 + (-16.0 - 16.0i)T + 169iT^{2} \) |
| 17 | \( 1 + (7.21 + 7.21i)T + 289iT^{2} \) |
| 19 | \( 1 - 8.06T + 361T^{2} \) |
| 23 | \( 1 + (-11.7 - 11.7i)T + 529iT^{2} \) |
| 29 | \( 1 - 6.17T + 841T^{2} \) |
| 31 | \( 1 - 41.4iT - 961T^{2} \) |
| 37 | \( 1 + (37.8 + 37.8i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 74.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (42.3 - 42.3i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-39.4 - 39.4i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (44.4 + 44.4i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 51.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 15.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (38.7 + 38.7i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 128. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (54.2 + 54.2i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 25.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-27.7 + 27.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 32.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (56.7 - 56.7i)T - 9.40e3iT^{2} \) |
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
−11.80549371782113552217145613463, −11.06024466356764328283906842042, −10.32295430196885279110624492370, −9.208441320375333025361747339870, −8.557264437891844712461487514471, −6.63330696874644365825363211849, −5.38631750637046158579747236100, −4.54355351376404580410407626065, −3.37704721907802563443498479324, −1.50965143082078983766904927878,
1.80523221726141990836538443545, 3.11762082102615599194149865824, 5.20311867062771621279371927833, 6.01298048794766947125884217246, 6.91057831086560845309748588132, 8.107609057231735136955665034441, 8.840734886553345752209642923974, 10.40605715482087025232212145858, 11.47177385841627021578097365640, 12.44278799795578878900868846238