L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.5i)9-s + (−0.5 − 0.866i)11-s + (1.22 + 1.22i)13-s + (−0.866 − 0.500i)14-s + (−0.5 + 0.866i)16-s + (0.965 + 0.258i)18-s + (0.707 + 0.707i)22-s + (−1.49 − 0.866i)26-s + (1.5 − 0.866i)31-s + (0.707 − 0.707i)43-s + 1.00i·49-s + 0.999·56-s + ⋯ |
L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.5i)9-s + (−0.5 − 0.866i)11-s + (1.22 + 1.22i)13-s + (−0.866 − 0.500i)14-s + (−0.5 + 0.866i)16-s + (0.965 + 0.258i)18-s + (0.707 + 0.707i)22-s + (−1.49 − 0.866i)26-s + (1.5 − 0.866i)31-s + (0.707 − 0.707i)43-s + 1.00i·49-s + 0.999·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6663987776\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6663987776\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 3 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 13 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 17 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 89 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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
−9.111409934785891487394802365445, −8.620345160379340901774229889217, −8.327560554515529022719506606111, −7.35183324109194090073564341983, −6.28317961894053315161225176589, −5.76066758841463795084346311750, −4.55761231543734524211270110022, −3.65844659407670652915571274315, −2.43833824575708422277377111812, −1.05749885022927163648463980764,
0.916510227195385905098943437994, 2.06765426917651471957245856196, 3.23349741532375650074172015590, 4.57978537725006966987753100316, 5.14378359628300897550399763439, 6.11653805523956191504989034023, 7.35821893994503716277185259360, 8.103570559006781653853434353207, 8.318362845604654385132785449155, 9.271432042953791997888296824423