L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 6·11-s − 12-s + 13-s + 16-s + 18-s + 6·19-s − 6·22-s + 6·23-s − 24-s + 26-s − 27-s + 2·29-s + 4·31-s + 32-s + 6·33-s + 36-s + 10·37-s + 6·38-s − 39-s − 6·41-s + 8·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.80·11-s − 0.288·12-s + 0.277·13-s + 1/4·16-s + 0.235·18-s + 1.37·19-s − 1.27·22-s + 1.25·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.176·32-s + 1.04·33-s + 1/6·36-s + 1.64·37-s + 0.973·38-s − 0.160·39-s − 0.937·41-s + 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.171628782\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.171628782\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.339196028527913011420132860253, −8.112828788704538337471455400291, −7.55791425352179681097047553232, −6.71090874096745784254140002420, −5.76561252871232307176139755211, −5.18274785794287017679754907894, −4.52351035255479102202576404027, −3.24832322846155318966060592343, −2.50735482069933531321125260906, −0.931742409450502916584918442470,
0.931742409450502916584918442470, 2.50735482069933531321125260906, 3.24832322846155318966060592343, 4.52351035255479102202576404027, 5.18274785794287017679754907894, 5.76561252871232307176139755211, 6.71090874096745784254140002420, 7.55791425352179681097047553232, 8.112828788704538337471455400291, 9.339196028527913011420132860253