| L(s) = 1 | + 2-s + 4-s − 5-s − 2·7-s + 8-s − 10-s + 11-s − 13-s − 2·14-s + 16-s + 4·17-s + 6·19-s − 20-s + 22-s + 6·23-s + 25-s − 26-s − 2·28-s − 8·29-s − 4·31-s + 32-s + 4·34-s + 2·35-s − 2·37-s + 6·38-s − 40-s + 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s − 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.970·17-s + 1.37·19-s − 0.223·20-s + 0.213·22-s + 1.25·23-s + 1/5·25-s − 0.196·26-s − 0.377·28-s − 1.48·29-s − 0.718·31-s + 0.176·32-s + 0.685·34-s + 0.338·35-s − 0.328·37-s + 0.973·38-s − 0.158·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.972317135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.972317135\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| 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
−16.18703159911553, −15.74904515358108, −15.05463271548257, −14.53468482011106, −14.17106368207217, −13.23018144789862, −13.02109390037242, −12.30042505436823, −11.85750163692149, −11.23972920662860, −10.69924735478045, −9.896343541707991, −9.353478664174234, −8.859165046310501, −7.669056416057577, −7.477030017218720, −6.840937383906377, −5.969960606606862, −5.477198882276331, −4.811258758347042, −3.925032383549919, −3.321872015931166, −2.887700018085402, −1.701620942724267, −0.7046804552682017,
0.7046804552682017, 1.701620942724267, 2.887700018085402, 3.321872015931166, 3.925032383549919, 4.811258758347042, 5.477198882276331, 5.969960606606862, 6.840937383906377, 7.477030017218720, 7.669056416057577, 8.859165046310501, 9.353478664174234, 9.896343541707991, 10.69924735478045, 11.23972920662860, 11.85750163692149, 12.30042505436823, 13.02109390037242, 13.23018144789862, 14.17106368207217, 14.53468482011106, 15.05463271548257, 15.74904515358108, 16.18703159911553
