| L(s) = 1 | + 3-s + 4·5-s − 7-s + 9-s − 2·11-s + 2·13-s + 4·15-s − 4·19-s − 21-s + 6·23-s + 11·25-s + 27-s + 10·29-s + 8·31-s − 2·33-s − 4·35-s − 10·37-s + 2·39-s − 4·41-s − 8·43-s + 4·45-s + 4·47-s + 49-s − 10·53-s − 8·55-s − 4·57-s + 8·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 1.03·15-s − 0.917·19-s − 0.218·21-s + 1.25·23-s + 11/5·25-s + 0.192·27-s + 1.85·29-s + 1.43·31-s − 0.348·33-s − 0.676·35-s − 1.64·37-s + 0.320·39-s − 0.624·41-s − 1.21·43-s + 0.596·45-s + 0.583·47-s + 1/7·49-s − 1.37·53-s − 1.07·55-s − 0.529·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.752319096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.752319096\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−9.714265809531109837764166879771, −8.713649165355041628416459255809, −8.391649339070399482373683841817, −6.83715734248313427167981756631, −6.48144389669749178167352152901, −5.44334341616198064453307723167, −4.66169216854101764844813417996, −3.16076221185606682123289148521, −2.45390467337760422022857720016, −1.34272176216502313926400283488,
1.34272176216502313926400283488, 2.45390467337760422022857720016, 3.16076221185606682123289148521, 4.66169216854101764844813417996, 5.44334341616198064453307723167, 6.48144389669749178167352152901, 6.83715734248313427167981756631, 8.391649339070399482373683841817, 8.713649165355041628416459255809, 9.714265809531109837764166879771
