L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 2·11-s + 13-s − 15-s − 4·17-s − 3·19-s − 21-s + 9·23-s − 4·25-s + 27-s − 29-s + 5·31-s + 2·33-s + 35-s − 8·37-s + 39-s + 6·41-s + 9·43-s − 45-s + 3·47-s + 49-s − 4·51-s + 3·53-s − 2·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.258·15-s − 0.970·17-s − 0.688·19-s − 0.218·21-s + 1.87·23-s − 4/5·25-s + 0.192·27-s − 0.185·29-s + 0.898·31-s + 0.348·33-s + 0.169·35-s − 1.31·37-s + 0.160·39-s + 0.937·41-s + 1.37·43-s − 0.149·45-s + 0.437·47-s + 1/7·49-s − 0.560·51-s + 0.412·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.087206603\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.087206603\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−8.485255636099615654192222300978, −7.65301309816855844423242627951, −6.90221224971830328003677088954, −6.41114902905702982840055909143, −5.37758607480975298731700050385, −4.36947028028289865835133717749, −3.85307545971783104916091823499, −2.94495441885535661130563296167, −2.06996964904661111806829340103, −0.791456854704856529223781893647,
0.791456854704856529223781893647, 2.06996964904661111806829340103, 2.94495441885535661130563296167, 3.85307545971783104916091823499, 4.36947028028289865835133717749, 5.37758607480975298731700050385, 6.41114902905702982840055909143, 6.90221224971830328003677088954, 7.65301309816855844423242627951, 8.485255636099615654192222300978