L(s) = 1 | − 2.41i·7-s − 5.41·11-s + 6.65·13-s + 3.41i·17-s − 7.24i·19-s + 2.58·23-s + 10.2i·29-s + 5.24i·31-s + 6.82·37-s + 4.82i·41-s + 3.58i·43-s − 7.41·47-s + 1.17·49-s − 0.828i·53-s + 5.07·59-s + ⋯ |
L(s) = 1 | − 0.912i·7-s − 1.63·11-s + 1.84·13-s + 0.828i·17-s − 1.66i·19-s + 0.539·23-s + 1.90i·29-s + 0.941i·31-s + 1.12·37-s + 0.754i·41-s + 0.546i·43-s − 1.08·47-s + 0.167·49-s − 0.113i·53-s + 0.660·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.892946591\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.892946591\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2.41iT - 7T^{2} \) |
| 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 - 6.65T + 13T^{2} \) |
| 17 | \( 1 - 3.41iT - 17T^{2} \) |
| 19 | \( 1 + 7.24iT - 19T^{2} \) |
| 23 | \( 1 - 2.58T + 23T^{2} \) |
| 29 | \( 1 - 10.2iT - 29T^{2} \) |
| 31 | \( 1 - 5.24iT - 31T^{2} \) |
| 37 | \( 1 - 6.82T + 37T^{2} \) |
| 41 | \( 1 - 4.82iT - 41T^{2} \) |
| 43 | \( 1 - 3.58iT - 43T^{2} \) |
| 47 | \( 1 + 7.41T + 47T^{2} \) |
| 53 | \( 1 + 0.828iT - 53T^{2} \) |
| 59 | \( 1 - 5.07T + 59T^{2} \) |
| 61 | \( 1 + 1.82T + 61T^{2} \) |
| 67 | \( 1 - 3.24iT - 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 16.4T + 73T^{2} \) |
| 79 | \( 1 + 12iT - 79T^{2} \) |
| 83 | \( 1 + 4.58T + 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 - 9T + 97T^{2} \) |
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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
−7.908965359466818797667200393683, −7.19366801281770301406378407084, −6.59369036203547291276378785460, −5.82343223346684487944331528592, −4.98349856849110734636748541898, −4.43226791724868203331082988113, −3.36792562242174965657467151721, −2.93660792828993910499531775638, −1.61379356915973691451341248411, −0.73391924644291854916889066924,
0.67920104162489992579883489351, 1.98450775448176746071945182712, 2.66538923387610898329670939853, 3.53887867411739176625892938354, 4.33059145868896031290073462302, 5.36198875534597758902456013450, 5.81060562612558195496032351338, 6.29104517983576202147799798215, 7.43650191707526471660655255000, 8.076415248459085086061857466111