L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 3·11-s − 13-s − 14-s + 16-s − 19-s + 20-s + 3·22-s − 6·23-s + 25-s + 26-s + 28-s − 6·29-s − 4·31-s − 32-s + 35-s + 2·37-s + 38-s − 40-s − 3·41-s − 43-s − 3·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.904·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.229·19-s + 0.223·20-s + 0.639·22-s − 1.25·23-s + 1/5·25-s + 0.196·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.169·35-s + 0.328·37-s + 0.162·38-s − 0.158·40-s − 0.468·41-s − 0.152·43-s − 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−8.842446145907377354309720673467, −8.010325691667264970342097497503, −7.51261249283968652756723628830, −6.50851058840715009838923325131, −5.67884583256951749386000854806, −4.90975129415303143434092118058, −3.68308733876239902645143187694, −2.47735725601765811027399642233, −1.67765329461785562945611255720, 0,
1.67765329461785562945611255720, 2.47735725601765811027399642233, 3.68308733876239902645143187694, 4.90975129415303143434092118058, 5.67884583256951749386000854806, 6.50851058840715009838923325131, 7.51261249283968652756723628830, 8.010325691667264970342097497503, 8.842446145907377354309720673467