L(s) = 1 | + 2-s − i·3-s − 4-s − i·6-s + 4i·7-s − 3·8-s − 9-s − 4i·11-s + i·12-s + 2·13-s + 4i·14-s − 16-s + (1 − 4i)17-s − 18-s − 4·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577i·3-s − 0.5·4-s − 0.408i·6-s + 1.51i·7-s − 1.06·8-s − 0.333·9-s − 1.20i·11-s + 0.288i·12-s + 0.554·13-s + 1.06i·14-s − 0.250·16-s + (0.242 − 0.970i)17-s − 0.235·18-s − 0.917·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.977239 - 0.120303i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.977239 - 0.120303i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 17 | \( 1 + (-1 + 4i)T \) |
good | 2 | \( 1 - T + 2T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 8iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 8iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 16iT - 97T^{2} \) |
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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
−15.22421906455026663818725423192, −14.20335462092373922512270459800, −13.22313622696751046955945037970, −12.29589703833658958394204491211, −11.29197683349063617608596857547, −9.170445946119745654646751668784, −8.397074917737072518802833061281, −6.25244206026251981780374239300, −5.25757798355882877965671943400, −3.08123160264227815861676586344,
3.83132277647941273193838504179, 4.72410306496764483411551904974, 6.58665318563334711739225264874, 8.353938731587443077985698253799, 9.868263186905898811129796321835, 10.77052143340783570038150502125, 12.48071732359980117615625341463, 13.37882699443855431166316779060, 14.47208214981668008954963250491, 15.20751821432646299185542006981