| L(s) = 1 | − 5-s + 2·7-s + 2·13-s + 3·17-s + 5·19-s − 3·23-s + 25-s + 6·29-s + 5·31-s − 2·35-s + 2·37-s − 12·41-s + 8·43-s + 12·47-s − 3·49-s + 3·53-s − 6·59-s − 7·61-s − 2·65-s + 2·67-s − 12·71-s − 16·73-s − 79-s + 15·83-s − 3·85-s + 12·89-s + 4·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s + 0.554·13-s + 0.727·17-s + 1.14·19-s − 0.625·23-s + 1/5·25-s + 1.11·29-s + 0.898·31-s − 0.338·35-s + 0.328·37-s − 1.87·41-s + 1.21·43-s + 1.75·47-s − 3/7·49-s + 0.412·53-s − 0.781·59-s − 0.896·61-s − 0.248·65-s + 0.244·67-s − 1.42·71-s − 1.87·73-s − 0.112·79-s + 1.64·83-s − 0.325·85-s + 1.27·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.524374499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.524374499\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−10.80799962022765933499638365454, −10.05815610462548757364234587037, −8.935925621137214438533796074475, −8.075757804585481763389308605687, −7.41769324058008963967501874807, −6.19290173727045335804565486285, −5.15656604704510166418064747088, −4.14163068051343696678482509679, −2.94641868410306170011925698041, −1.24500108130507230710940864754,
1.24500108130507230710940864754, 2.94641868410306170011925698041, 4.14163068051343696678482509679, 5.15656604704510166418064747088, 6.19290173727045335804565486285, 7.41769324058008963967501874807, 8.075757804585481763389308605687, 8.935925621137214438533796074475, 10.05815610462548757364234587037, 10.80799962022765933499638365454
