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 + (−5.12 + 4.76i)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 + (−0.732 + 0.680i)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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 - 0.463i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.886 - 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.144516 + 0.587843i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.144516 + 0.587843i\) |
\(L(2)\) |
|
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 | \( 1 + (5.12 - 4.76i)T \) |
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
−11.76505053985627585587972395199, −10.97110459039933599754093600321, −9.788471007175129680986427147741, −8.381381179474361392789471059727, −7.45653458200329815622515726594, −6.31319289016358278483357937092, −5.33074779047813202843041380541, −3.43511082789597758592839356470, −2.53723898198945186121557153291, −0.27545523773686160388288353042,
3.16827323255104052313015191819, 4.39059699588372946273741236339, 4.89970153610569940883116961204, 6.61999143653573221798297513250, 7.50074850304284422919309982529, 8.835039777640939507584250887525, 9.671226622729182655057656554040, 10.75908390047209720849187020180, 11.97577249442753004091930155916, 12.49754489202524939523234352408