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 | \( 1 + (56.7 + 56.7i)T + 9.40e3iT^{2} \) |
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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
−12.44278799795578878900868846238, −11.47177385841627021578097365640, −10.40605715482087025232212145858, −8.840734886553345752209642923974, −8.107609057231735136955665034441, −6.91057831086560845309748588132, −6.01298048794766947125884217246, −5.20311867062771621279371927833, −3.11762082102615599194149865824, −1.80523221726141990836538443545,
1.50965143082078983766904927878, 3.37704721907802563443498479324, 4.54355351376404580410407626065, 5.38631750637046158579747236100, 6.63330696874644365825363211849, 8.557264437891844712461487514471, 9.208441320375333025361747339870, 10.32295430196885279110624492370, 11.06024466356764328283906842042, 11.80549371782113552217145613463