| L(s) = 1 | + (−1.36 − 0.366i)2-s + (1.73 + i)4-s − 2.73i·7-s + (−1.99 − 2i)8-s − 2i·11-s − 3.46·13-s + (−1 + 3.73i)14-s + (1.99 + 3.46i)16-s + 3.46i·17-s + 7.46i·19-s + (−0.732 + 2.73i)22-s + 4.19i·23-s + (4.73 + 1.26i)26-s + (2.73 − 4.73i)28-s + 6.92i·29-s + ⋯ |
| L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.5i)4-s − 1.03i·7-s + (−0.707 − 0.707i)8-s − 0.603i·11-s − 0.960·13-s + (−0.267 + 0.997i)14-s + (0.499 + 0.866i)16-s + 0.840i·17-s + 1.71i·19-s + (−0.156 + 0.582i)22-s + 0.874i·23-s + (0.928 + 0.248i)26-s + (0.516 − 0.894i)28-s + 1.28i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9136591833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9136591833\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.36 + 0.366i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2.73iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 7.46iT - 19T^{2} \) |
| 23 | \( 1 - 4.19iT - 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 5.46T + 41T^{2} \) |
| 43 | \( 1 - 8.73T + 43T^{2} \) |
| 47 | \( 1 + 6.73iT - 47T^{2} \) |
| 53 | \( 1 - 4.53T + 53T^{2} \) |
| 59 | \( 1 - 0.535iT - 59T^{2} \) |
| 61 | \( 1 + 4.92iT - 61T^{2} \) |
| 67 | \( 1 - 7.26T + 67T^{2} \) |
| 71 | \( 1 - 1.46T + 71T^{2} \) |
| 73 | \( 1 + 0.535iT - 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 + 4.92T + 89T^{2} \) |
| 97 | \( 1 - 6.39iT - 97T^{2} \) |
| 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
−9.401011899761475155173877796076, −8.503784013105498269242237711225, −7.76676941019786579558448310724, −7.24609792047909314652850596483, −6.32290512061017416769039411460, −5.42194278750392307123297804497, −3.98609820394365710643385744079, −3.39152540294231288304865650048, −2.04677501806260504717959086173, −0.957815159753861520918419868526,
0.58206049848780155598991877222, 2.44817841408723232151916021635, 2.55100191328753643492725538487, 4.52498286377547978460397197121, 5.28614684557470177210932685509, 6.21577791582568621600377915153, 7.03727196670097998134679524026, 7.64820642608181591344972851501, 8.575623260239368178825318422656, 9.341333689431807077135299696962
