| L(s) = 1 | − 2i·2-s − 3i·3-s − 4·4-s − 6·6-s − 7i·7-s + 8i·8-s − 9·9-s − 72·11-s + 12i·12-s − 34i·13-s − 14·14-s + 16·16-s − 6i·17-s + 18i·18-s − 92·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.377i·7-s + 0.353i·8-s − 0.333·9-s − 1.97·11-s + 0.288i·12-s − 0.725i·13-s − 0.267·14-s + 0.250·16-s − 0.0856i·17-s + 0.235i·18-s − 1.11·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3205477649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3205477649\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2iT \) |
| 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7iT \) |
| good | 11 | \( 1 + 72T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 6iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 92T + 6.85e3T^{2} \) |
| 23 | \( 1 + 180iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 114T + 2.43e4T^{2} \) |
| 31 | \( 1 - 56T + 2.97e4T^{2} \) |
| 37 | \( 1 - 34iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 164iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 168iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 654iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 492T + 2.05e5T^{2} \) |
| 61 | \( 1 + 250T + 2.26e5T^{2} \) |
| 67 | \( 1 - 124iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 36T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.01e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 56T + 4.93e5T^{2} \) |
| 83 | \( 1 - 228iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 390T + 7.04e5T^{2} \) |
| 97 | \( 1 - 70iT - 9.12e5T^{2} \) |
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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
−9.902572049413084105423211540504, −8.491964011067177279315270661250, −8.182550628865246096383110765188, −7.21373118182298356981701827252, −6.15181513939298232046157140951, −5.16973699601841781779310227131, −4.31735279856206204945564502214, −2.87732912128440471104072197174, −2.36795900121303239473496621988, −0.834321493378490816112253835002,
0.099657920157259991319029215947, 2.10414078823841281204173772513, 3.26368182305083798240671594322, 4.44718156564669562171139120473, 5.21577304542643801661899534120, 5.91849451028325975658258516513, 6.97030323722475017830532386355, 7.930031724357838386746408102899, 8.512656660960623474220395965697, 9.427068687014456601852482409341
