| L(s) = 1 | − 3.31i·2-s − 7·4-s + 9.38·5-s − 9.89i·7-s + 9.94i·8-s − 31.1i·10-s − 12.7i·13-s − 32.8·14-s + 5.00·16-s + 4.24i·19-s − 65.6·20-s + 28.1·23-s + 63·25-s − 42.2·26-s + 69.2i·28-s − 13.2i·29-s + ⋯ |
| L(s) = 1 | − 1.65i·2-s − 1.75·4-s + 1.87·5-s − 1.41i·7-s + 1.24i·8-s − 3.11i·10-s − 0.979i·13-s − 2.34·14-s + 0.312·16-s + 0.223i·19-s − 3.28·20-s + 1.22·23-s + 2.52·25-s − 1.62·26-s + 2.47i·28-s − 0.457i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.399310895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.399310895\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 3.31iT - 4T^{2} \) |
| 5 | \( 1 - 9.38T + 25T^{2} \) |
| 7 | \( 1 + 9.89iT - 49T^{2} \) |
| 13 | \( 1 + 12.7iT - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 4.24iT - 361T^{2} \) |
| 23 | \( 1 - 28.1T + 529T^{2} \) |
| 29 | \( 1 + 13.2iT - 841T^{2} \) |
| 31 | \( 1 + 44T + 961T^{2} \) |
| 37 | \( 1 - 44T + 1.36e3T^{2} \) |
| 41 | \( 1 + 39.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 29.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 9.38T + 2.20e3T^{2} \) |
| 53 | \( 1 + 65.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 18.7T + 3.48e3T^{2} \) |
| 61 | \( 1 + 69.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 88T + 4.48e3T^{2} \) |
| 71 | \( 1 + 46.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 41.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 12.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 92.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 112.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 70T + 9.40e3T^{2} \) |
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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
−9.575878218797367723611088638797, −8.953136996393498485030095026338, −7.62580186979246067903255259567, −6.56920063318945037853820620513, −5.51852723810499299525545315646, −4.64814669274223822233158574322, −3.51089818105433550452126976807, −2.61983034328304331452748967319, −1.56903068604804080264196699389, −0.74561151750128686202823466109,
1.72826571302154850162004140093, 2.76578377136109493261548190997, 4.70550019212804737248923362077, 5.37003515282712208716911551540, 6.00977486030524172566782069196, 6.54277218688418515081979300022, 7.42012829893964632905607801703, 8.744479313584144876238978895618, 9.103921021128556202077558583317, 9.518415193119666147262255214238
