| L(s) = 1 | + (4.36 + 2.44i)5-s − 2.79i·7-s + 18.1i·11-s − 23.0i·13-s − 5.72·17-s − 23.1·19-s + 0.271·23-s + (13.0 + 21.2i)25-s + 39.7i·29-s − 47.3·31-s + (6.82 − 12.2i)35-s − 34.8i·37-s + 13.2i·41-s + 46.7i·43-s − 40.9·47-s + ⋯ |
| L(s) = 1 | + (0.872 + 0.488i)5-s − 0.399i·7-s + 1.64i·11-s − 1.77i·13-s − 0.336·17-s − 1.22·19-s + 0.0118·23-s + (0.523 + 0.851i)25-s + 1.37i·29-s − 1.52·31-s + (0.194 − 0.348i)35-s − 0.942i·37-s + 0.323i·41-s + 1.08i·43-s − 0.870·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8496788225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8496788225\) |
| \(L(2)\) |
|
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 | \( 1 + (-4.36 - 2.44i)T \) |
| good | 7 | \( 1 + 2.79iT - 49T^{2} \) |
| 11 | \( 1 - 18.1iT - 121T^{2} \) |
| 13 | \( 1 + 23.0iT - 169T^{2} \) |
| 17 | \( 1 + 5.72T + 289T^{2} \) |
| 19 | \( 1 + 23.1T + 361T^{2} \) |
| 23 | \( 1 - 0.271T + 529T^{2} \) |
| 29 | \( 1 - 39.7iT - 841T^{2} \) |
| 31 | \( 1 + 47.3T + 961T^{2} \) |
| 37 | \( 1 + 34.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 13.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 46.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 40.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 91.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 78.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 31.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 6.91iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 81.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 106. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 63.5T + 6.24e3T^{2} \) |
| 83 | \( 1 + 0.284T + 6.88e3T^{2} \) |
| 89 | \( 1 - 28.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 92.9iT - 9.40e3T^{2} \) |
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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
−9.387693470184420087427558469301, −8.458272995796831702604062029001, −7.45469817745950880549964314735, −7.01166306414841305364930543252, −6.07213391360948891186447392939, −5.28446453240214151900805790271, −4.46434148931892996586207064854, −3.33450483032914835040803969372, −2.37766732125980999550645154237, −1.47463515624593381083772208146,
0.18879155904879151652914180847, 1.65543679345467919426987487368, 2.37979568795128069548292038516, 3.66835289118221716613950145951, 4.55443008358671759128392437993, 5.49380056410662025556531619293, 6.25463217206279760292280676176, 6.68297579499378639620142123779, 8.036487796733396788376215275443, 8.808422448399783188350318927525
