L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s + 3·11-s + 6·13-s + 2·14-s + 16-s − 2·17-s − 19-s + 3·22-s − 23-s + 6·26-s + 2·28-s + 5·29-s + 7·31-s + 32-s − 2·34-s + 2·37-s − 38-s − 2·41-s + 6·43-s + 3·44-s − 46-s − 12·47-s − 3·49-s + 6·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s + 0.904·11-s + 1.66·13-s + 0.534·14-s + 1/4·16-s − 0.485·17-s − 0.229·19-s + 0.639·22-s − 0.208·23-s + 1.17·26-s + 0.377·28-s + 0.928·29-s + 1.25·31-s + 0.176·32-s − 0.342·34-s + 0.328·37-s − 0.162·38-s − 0.312·41-s + 0.914·43-s + 0.452·44-s − 0.147·46-s − 1.75·47-s − 3/7·49-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.570560019\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.570560019\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−7.80804397266607189559216033064, −6.83293867677472438024126391093, −6.33700991371991327403869549397, −5.81425238052656344069247028792, −4.81611712183270306578535924149, −4.29897823442123687058667791218, −3.64282975666973200040110340921, −2.77185220154074539666627057035, −1.72475241995813216950786621807, −1.03553655204268089139986664994,
1.03553655204268089139986664994, 1.72475241995813216950786621807, 2.77185220154074539666627057035, 3.64282975666973200040110340921, 4.29897823442123687058667791218, 4.81611712183270306578535924149, 5.81425238052656344069247028792, 6.33700991371991327403869549397, 6.83293867677472438024126391093, 7.80804397266607189559216033064