| L(s) = 1 | − 6.92i·7-s + 20.7i·11-s + 14·13-s − 6·17-s − 6.92i·19-s − 30·29-s + 20.7i·31-s − 26·37-s + 54·41-s + 20.7i·43-s + 41.5i·47-s + 1.00·49-s − 18·53-s − 20.7i·59-s − 70·61-s + ⋯ |
| L(s) = 1 | − 0.989i·7-s + 1.88i·11-s + 1.07·13-s − 0.352·17-s − 0.364i·19-s − 1.03·29-s + 0.670i·31-s − 0.702·37-s + 1.31·41-s + 0.483i·43-s + 0.884i·47-s + 0.0204·49-s − 0.339·53-s − 0.352i·59-s − 1.14·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.104504054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.104504054\) |
| \(L(2)\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 6.92iT - 49T^{2} \) |
| 11 | \( 1 - 20.7iT - 121T^{2} \) |
| 13 | \( 1 - 14T + 169T^{2} \) |
| 17 | \( 1 + 6T + 289T^{2} \) |
| 19 | \( 1 + 6.92iT - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 30T + 841T^{2} \) |
| 31 | \( 1 - 20.7iT - 961T^{2} \) |
| 37 | \( 1 + 26T + 1.36e3T^{2} \) |
| 41 | \( 1 - 54T + 1.68e3T^{2} \) |
| 43 | \( 1 - 20.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 41.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 18T + 2.80e3T^{2} \) |
| 59 | \( 1 + 20.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 70T + 3.72e3T^{2} \) |
| 67 | \( 1 + 117. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 83.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 82T + 5.32e3T^{2} \) |
| 79 | \( 1 - 76.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 20.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 114T + 7.92e3T^{2} \) |
| 97 | \( 1 + 34T + 9.40e3T^{2} \) |
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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
−8.640319558997984224862061876454, −7.69396684445018309300259570279, −7.19296354124681964389049195463, −6.57988998958734997298936193586, −5.63510139712762159687404127870, −4.56469614363339078130256644831, −4.20434195282538967110093625526, −3.20112353829536528944775161019, −2.00316498352535072697552514832, −1.17601034565321495782504936832,
0.24298753165683321568356317817, 1.44060640897852131205676832981, 2.56831561782708642830384265904, 3.40748014142372333258362601166, 4.11130257710220584889673665459, 5.43992601337769134409495725624, 5.83612839677513949501907468566, 6.38752857374204609479597591773, 7.51684889118965450897388060599, 8.319786524871782816745601057951
