| L(s) = 1 | + 2.21i·3-s + i·5-s − 4.12·7-s − 1.92·9-s + 3.11·11-s − 6.05·13-s − 2.21·15-s − 4.63i·17-s + 8.34·19-s − 9.14i·21-s + (1.87 − 4.41i)23-s − 25-s + 2.38i·27-s + 5.98·29-s − 6.26i·31-s + ⋯ |
| L(s) = 1 | + 1.28i·3-s + 0.447i·5-s − 1.55·7-s − 0.642·9-s + 0.940·11-s − 1.67·13-s − 0.573·15-s − 1.12i·17-s + 1.91·19-s − 1.99i·21-s + (0.391 − 0.920i)23-s − 0.200·25-s + 0.458i·27-s + 1.11·29-s − 1.12i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.274297969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.274297969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-1.87 + 4.41i)T \) |
| good | 3 | \( 1 - 2.21iT - 3T^{2} \) |
| 7 | \( 1 + 4.12T + 7T^{2} \) |
| 11 | \( 1 - 3.11T + 11T^{2} \) |
| 13 | \( 1 + 6.05T + 13T^{2} \) |
| 17 | \( 1 + 4.63iT - 17T^{2} \) |
| 19 | \( 1 - 8.34T + 19T^{2} \) |
| 29 | \( 1 - 5.98T + 29T^{2} \) |
| 31 | \( 1 + 6.26iT - 31T^{2} \) |
| 37 | \( 1 - 0.450iT - 37T^{2} \) |
| 41 | \( 1 + 9.08T + 41T^{2} \) |
| 43 | \( 1 + 8.68T + 43T^{2} \) |
| 47 | \( 1 + 9.51iT - 47T^{2} \) |
| 53 | \( 1 - 3.79iT - 53T^{2} \) |
| 59 | \( 1 + 6.08iT - 59T^{2} \) |
| 61 | \( 1 + 6.04iT - 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 6.47iT - 71T^{2} \) |
| 73 | \( 1 - 9.18T + 73T^{2} \) |
| 79 | \( 1 - 8.25T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 4.49iT - 89T^{2} \) |
| 97 | \( 1 - 15.2iT - 97T^{2} \) |
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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
−8.985685781720906459632644047178, −7.75979078890047660215003024320, −6.75468931764253880589828401681, −6.68926999422952780047103723327, −5.23690384620209549171617008442, −4.91041541652506291329689066118, −3.75592588661233103614620880445, −3.23561942704479312082737779113, −2.49880591252941137043119822607, −0.49341629877066163815587254145,
0.872101657303368990733926533955, 1.73721363101553654561776854413, 2.94307536258446577316355021335, 3.56058245912350027074841185445, 4.83031742666694398721202292008, 5.63430833137172587793511317975, 6.63184940022245474873068345355, 6.82969804259135776800807394106, 7.59636826524388463188004843588, 8.347594561038831008465934766946
