L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 8-s + 9-s − 2·10-s − 12-s + 2·13-s − 2·15-s + 16-s − 18-s − 4·19-s + 2·20-s + 24-s − 25-s − 2·26-s − 27-s − 10·29-s + 2·30-s − 8·31-s − 32-s + 36-s + 2·37-s + 4·38-s − 2·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s + 0.554·13-s − 0.516·15-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.204·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s − 1.85·29-s + 0.365·30-s − 1.43·31-s − 0.176·32-s + 1/6·36-s + 0.328·37-s + 0.648·38-s − 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209814 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209814 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5812047095\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5812047095\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.04692397817100, −12.67056541084669, −11.92037757932040, −11.46800653910542, −11.09147213900404, −10.63897328501242, −10.27331749715832, −9.532650472038098, −9.498198523780816, −8.803892962334295, −8.429434660790663, −7.666645740517029, −7.382098903943955, −6.723939362760841, −6.247924841075336, −5.842628703366251, −5.457888889292154, −4.876616501677762, −4.030963345070786, −3.710675047313068, −2.885944476613487, −2.133445133830167, −1.766963879451072, −1.232211170160349, −0.2473208441737513,
0.2473208441737513, 1.232211170160349, 1.766963879451072, 2.133445133830167, 2.885944476613487, 3.710675047313068, 4.030963345070786, 4.876616501677762, 5.457888889292154, 5.842628703366251, 6.247924841075336, 6.723939362760841, 7.382098903943955, 7.666645740517029, 8.429434660790663, 8.803892962334295, 9.498198523780816, 9.532650472038098, 10.27331749715832, 10.63897328501242, 11.09147213900404, 11.46800653910542, 11.92037757932040, 12.67056541084669, 13.04692397817100