L(s) = 1 | − 1.41·2-s − 3-s − 5-s + 1.41·6-s − 7-s + 2.82·8-s + 9-s + 1.41·10-s + 11-s + 0.585·13-s + 1.41·14-s + 15-s − 4.00·16-s − 1.82·17-s − 1.41·18-s − 3.82·19-s + 21-s − 1.41·22-s − 3.82·23-s − 2.82·24-s + 25-s − 0.828·26-s − 27-s − 1.24·29-s − 1.41·30-s + ⋯ |
L(s) = 1 | − 1.00·2-s − 0.577·3-s − 0.447·5-s + 0.577·6-s − 0.377·7-s + 0.999·8-s + 0.333·9-s + 0.447·10-s + 0.301·11-s + 0.162·13-s + 0.377·14-s + 0.258·15-s − 1.00·16-s − 0.443·17-s − 0.333·18-s − 0.878·19-s + 0.218·21-s − 0.301·22-s − 0.798·23-s − 0.577·24-s + 0.200·25-s − 0.162·26-s − 0.192·27-s − 0.230·29-s − 0.258·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4628240004\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4628240004\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 13 | \( 1 - 0.585T + 13T^{2} \) |
| 17 | \( 1 + 1.82T + 17T^{2} \) |
| 19 | \( 1 + 3.82T + 19T^{2} \) |
| 23 | \( 1 + 3.82T + 23T^{2} \) |
| 29 | \( 1 + 1.24T + 29T^{2} \) |
| 31 | \( 1 - 0.585T + 31T^{2} \) |
| 37 | \( 1 + 6.24T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 6.41T + 43T^{2} \) |
| 47 | \( 1 - 1.75T + 47T^{2} \) |
| 53 | \( 1 - 8.65T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 6T + 67T^{2} \) |
| 71 | \( 1 - 3.07T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 8.24T + 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 + 0.0710T + 89T^{2} \) |
| 97 | \( 1 + 7.24T + 97T^{2} \) |
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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
−9.852461270873464615123075745317, −8.897186008981605879157230099041, −8.368191384666051087933540030508, −7.37279330383101530119500603609, −6.68764975891301449682139812426, −5.66873358087859987248356378224, −4.50860217707260220428866815730, −3.78998316776499646749278114644, −2.06472346318772810908291847475, −0.60723918796020736205255592239,
0.60723918796020736205255592239, 2.06472346318772810908291847475, 3.78998316776499646749278114644, 4.50860217707260220428866815730, 5.66873358087859987248356378224, 6.68764975891301449682139812426, 7.37279330383101530119500603609, 8.368191384666051087933540030508, 8.897186008981605879157230099041, 9.852461270873464615123075745317