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 | \( 1 + 6.39iT - 97T^{2} \) |
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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
−9.341333689431807077135299696962, −8.575623260239368178825318422656, −7.64820642608181591344972851501, −7.03727196670097998134679524026, −6.21577791582568621600377915153, −5.28614684557470177210932685509, −4.52498286377547978460397197121, −2.55100191328753643492725538487, −2.44817841408723232151916021635, −0.58206049848780155598991877222,
0.957815159753861520918419868526, 2.04677501806260504717959086173, 3.39152540294231288304865650048, 3.98609820394365710643385744079, 5.42194278750392307123297804497, 6.32290512061017416769039411460, 7.24609792047909314652850596483, 7.76676941019786579558448310724, 8.503784013105498269242237711225, 9.401011899761475155173877796076