| L(s) = 1 | − 1.03i·2-s + 2.92·4-s − 1.46·5-s − 0.517i·7-s − 7.17i·8-s + 1.51i·10-s + 18.9i·13-s − 0.535·14-s + 4.28·16-s + 18.2i·17-s + 17.3i·19-s − 4.28·20-s − 40.6·23-s − 22.8·25-s + 19.6·26-s + ⋯ |
| L(s) = 1 | − 0.517i·2-s + 0.732·4-s − 0.292·5-s − 0.0739i·7-s − 0.896i·8-s + 0.151i·10-s + 1.45i·13-s − 0.0382·14-s + 0.267·16-s + 1.07i·17-s + 0.915i·19-s − 0.214·20-s − 1.76·23-s − 0.914·25-s + 0.754·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.611697997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.611697997\) |
| \(L(2)\) |
|
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 | \( 1 + 1.03iT - 4T^{2} \) |
| 5 | \( 1 + 1.46T + 25T^{2} \) |
| 7 | \( 1 + 0.517iT - 49T^{2} \) |
| 13 | \( 1 - 18.9iT - 169T^{2} \) |
| 17 | \( 1 - 18.2iT - 289T^{2} \) |
| 19 | \( 1 - 17.3iT - 361T^{2} \) |
| 23 | \( 1 + 40.6T + 529T^{2} \) |
| 29 | \( 1 - 43.0iT - 841T^{2} \) |
| 31 | \( 1 - 29.7T + 961T^{2} \) |
| 37 | \( 1 + 30.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 0.480iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 30.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 71.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 45.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 33.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 70.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 79.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 74.2T + 5.04e3T^{2} \) |
| 73 | \( 1 - 2.06iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 17.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 48.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 56.7T + 7.92e3T^{2} \) |
| 97 | \( 1 + 138.T + 9.40e3T^{2} \) |
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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
−10.12723398449220408425880721247, −9.055566662329138041224959683254, −8.136990912200828125565064863187, −7.28798026101611736239361267250, −6.45660493008705657432841958935, −5.72375915606075983748954179142, −4.14591216911879078774303876576, −3.69370510571074994563958899804, −2.24133162149808805664573584350, −1.47165042827806192874556241709,
0.45919069746965776049783408575, 2.21738633539039472129397783641, 3.07440597806449327624090637901, 4.38035061555196622859593323936, 5.51996743432892389448272729943, 6.10300704335667886734539125592, 7.13953887597144912283563417857, 7.84382761505792797392017883728, 8.351444784971471191277749364370, 9.627035533412801587204470953778
