| L(s) = 1 | − 1.41i·2-s + 2.41i·3-s + 3.41·6-s − 2.82i·8-s − 2.82·9-s − 5.82·11-s + 1.58i·13-s − 4.00·16-s + 5.24i·17-s + 4i·18-s − 6·19-s + 8.24i·22-s + 4.58i·23-s + 6.82·24-s + 2.24·26-s + 0.414i·27-s + ⋯ |
| L(s) = 1 | − 0.999i·2-s + 1.39i·3-s + 1.39·6-s − 0.999i·8-s − 0.942·9-s − 1.75·11-s + 0.439i·13-s − 1.00·16-s + 1.27i·17-s + 0.942i·18-s − 1.37·19-s + 1.75i·22-s + 0.956i·23-s + 1.39·24-s + 0.439·26-s + 0.0797i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8110574785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8110574785\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.41iT - 2T^{2} \) |
| 3 | \( 1 - 2.41iT - 3T^{2} \) |
| 11 | \( 1 + 5.82T + 11T^{2} \) |
| 13 | \( 1 - 1.58iT - 13T^{2} \) |
| 17 | \( 1 - 5.24iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 4.58iT - 23T^{2} \) |
| 29 | \( 1 + 2.65T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 - 6.24iT - 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 1.24iT - 47T^{2} \) |
| 53 | \( 1 + 4.24iT - 53T^{2} \) |
| 59 | \( 1 + 6.24T + 59T^{2} \) |
| 61 | \( 1 - 2.82T + 61T^{2} \) |
| 67 | \( 1 + 0.242iT - 67T^{2} \) |
| 71 | \( 1 + 8.82T + 71T^{2} \) |
| 73 | \( 1 - 8.48iT - 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 + 4.75iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.16267491440172677444891932462, −9.601291724373722324816219953672, −8.620502204500159371870507338302, −7.76037226018954255244563242236, −6.53326402621313878751977596704, −5.51870060720393095539463448609, −4.54104191858632295215648241150, −3.81586321905988346677616454748, −2.91641999633163643795672602249, −1.87015858748459871769252328118,
0.30563185975764321630357699312, 2.17246388166365842908076329541, 2.72720781824124058285194022912, 4.69419364329223309662316416713, 5.56780775405479700421169115324, 6.30625124040644672737329991808, 7.07473151241995711956821155016, 7.72207077803613579800226696945, 8.125257312603343690969921542023, 9.048932867088024778413584334170
