| L(s) = 1 | − 1.41·2-s − 1.73i·3-s + 2.00·4-s + 2.44i·6-s + (1.57 − 6.81i)7-s − 2.82·8-s − 2.99·9-s − 8.15·11-s − 3.46i·12-s + 14.6i·13-s + (−2.23 + 9.64i)14-s + 4.00·16-s − 5.81i·17-s + 4.24·18-s + 33.7i·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577i·3-s + 0.500·4-s + 0.408i·6-s + (0.225 − 0.974i)7-s − 0.353·8-s − 0.333·9-s − 0.741·11-s − 0.288i·12-s + 1.12i·13-s + (−0.159 + 0.688i)14-s + 0.250·16-s − 0.342i·17-s + 0.235·18-s + 1.77i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.248738644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.248738644\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.57 + 6.81i)T \) |
| good | 11 | \( 1 + 8.15T + 121T^{2} \) |
| 13 | \( 1 - 14.6iT - 169T^{2} \) |
| 17 | \( 1 + 5.81iT - 289T^{2} \) |
| 19 | \( 1 - 33.7iT - 361T^{2} \) |
| 23 | \( 1 - 37.2T + 529T^{2} \) |
| 29 | \( 1 - 9.25T + 841T^{2} \) |
| 31 | \( 1 - 19.2iT - 961T^{2} \) |
| 37 | \( 1 - 63.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 8.25iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 42.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 23.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 71.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 42.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 34.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.99T + 4.48e3T^{2} \) |
| 71 | \( 1 + 38.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 124. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 56.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 90.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 16.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 82.7iT - 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.758541493860087183504567979195, −8.730909155024972759037991269890, −8.002668944848348010457321476937, −7.23724268212017451050999550083, −6.67361203338321319577779388980, −5.54721896378215602601832475819, −4.39741494714458217752159462065, −3.17652825376759235271583297887, −1.89131506436671914807147648220, −0.868863768381750448710895648200,
0.67656263667089504789955062196, 2.48183225547135427162226815414, 3.06543415914886921764881730322, 4.72813818611067961592157135694, 5.39922650020626059921536454143, 6.34943167219175801158298434607, 7.48481922677759637152537749980, 8.272862072391690467815373159310, 8.985883953199646595096216504227, 9.611211076641843680958109736487
