L(s) = 1 | − 0.864i·7-s + 3.90·11-s + 1.13·13-s − 3.71i·17-s + 1.72i·19-s + 9.03·23-s − 1.26i·29-s + 3.25i·31-s + 6.38·37-s + 6.39i·41-s + 4.77i·43-s − 4.59·47-s + 6.25·49-s + 8.98i·53-s − 8.50·59-s + ⋯ |
L(s) = 1 | − 0.326i·7-s + 1.17·11-s + 0.314·13-s − 0.900i·17-s + 0.396i·19-s + 1.88·23-s − 0.235i·29-s + 0.584i·31-s + 1.05·37-s + 0.998i·41-s + 0.728i·43-s − 0.670·47-s + 0.893·49-s + 1.23i·53-s − 1.10·59-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
\(L(1)\) |
\(\approx\) |
\(2.404974473\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.404974473\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.864iT - 7T^{2} \) |
| 11 | \( 1 - 3.90T + 11T^{2} \) |
| 13 | \( 1 - 1.13T + 13T^{2} \) |
| 17 | \( 1 + 3.71iT - 17T^{2} \) |
| 19 | \( 1 - 1.72iT - 19T^{2} \) |
| 23 | \( 1 - 9.03T + 23T^{2} \) |
| 29 | \( 1 + 1.26iT - 29T^{2} \) |
| 31 | \( 1 - 3.25iT - 31T^{2} \) |
| 37 | \( 1 - 6.38T + 37T^{2} \) |
| 41 | \( 1 - 6.39iT - 41T^{2} \) |
| 43 | \( 1 - 4.77iT - 43T^{2} \) |
| 47 | \( 1 + 4.59T + 47T^{2} \) |
| 53 | \( 1 - 8.98iT - 53T^{2} \) |
| 59 | \( 1 + 8.50T + 59T^{2} \) |
| 61 | \( 1 + 9.04T + 61T^{2} \) |
| 67 | \( 1 + 11.0iT - 67T^{2} \) |
| 71 | \( 1 + 8.10T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 + 14.2iT - 79T^{2} \) |
| 83 | \( 1 - 8.10T + 83T^{2} \) |
| 89 | \( 1 + 3.56iT - 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.72533288485868182370066040340, −7.28753044016099497720230623724, −6.43244906370162764964953010229, −6.02362371191419407025157172513, −4.83999451260727822828971858013, −4.49493346058789460247944738836, −3.41765140174528458986790871212, −2.89013679605357945537880425349, −1.57083226138984972425525395903, −0.834845879478077752826940407930,
0.844126888355816544380332538854, 1.75483872873166231458552793326, 2.77158899330538585663484540649, 3.63462525427645267274563723763, 4.28664313470383433673666654965, 5.15651044459697362125410673445, 5.89395794322934655258536028212, 6.59095402148912456140625929484, 7.11972581898316141305677458283, 7.982978687693812555593599360259