| L(s) = 1 | − 1.41i·2-s − 2.00·4-s − 6.87i·5-s − 5.43·7-s + 2.82i·8-s − 9.72·10-s − 7.68i·11-s − 5.83·13-s + 7.68i·14-s + 4.00·16-s + 17.8i·17-s − 10.8·19-s + 13.7i·20-s − 10.8·22-s + 4.79i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 1.37i·5-s − 0.776·7-s + 0.353i·8-s − 0.972·10-s − 0.699i·11-s − 0.449·13-s + 0.549i·14-s + 0.250·16-s + 1.04i·17-s − 0.569·19-s + 0.687i·20-s − 0.494·22-s + 0.208i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.199395 + 0.385201i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.199395 + 0.385201i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 23 | \( 1 - 4.79iT \) |
| good | 5 | \( 1 + 6.87iT - 25T^{2} \) |
| 7 | \( 1 + 5.43T + 49T^{2} \) |
| 11 | \( 1 + 7.68iT - 121T^{2} \) |
| 13 | \( 1 + 5.83T + 169T^{2} \) |
| 17 | \( 1 - 17.8iT - 289T^{2} \) |
| 19 | \( 1 + 10.8T + 361T^{2} \) |
| 29 | \( 1 + 4.08iT - 841T^{2} \) |
| 31 | \( 1 + 16.2T + 961T^{2} \) |
| 37 | \( 1 + 5.67T + 1.36e3T^{2} \) |
| 41 | \( 1 - 54.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 36.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 46.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 83.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 4.39T + 3.72e3T^{2} \) |
| 67 | \( 1 - 37.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 30.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 141.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 69.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 69.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 67.1T + 9.40e3T^{2} \) |
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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
−10.35465779374558238752070638851, −9.524547388111103894435971056397, −8.728192076520571859860353174209, −8.024989113004889788012841871389, −6.43475286464423142305289222646, −5.38837900729138920310345789540, −4.35609803255014749286135228694, −3.25986905942932087120582466472, −1.63796446321219872776164310853, −0.18169444340751160721037315628,
2.48576178617815815274286109356, 3.60350799268433108658636731029, 4.95634790197963845405084588584, 6.22569930893118532433837377987, 6.97054633191247102972493598935, 7.49737920113484762208850842192, 8.891511868309359261459796710861, 9.849914304205782804754579860832, 10.44454876515528884451852042242, 11.55553106253942771576232698103
