L(s) = 1 | + 1.52·3-s + 1.39·7-s − 0.662·9-s + 11-s − 6.05·13-s − 3.26·17-s − 19-s + 2.13·21-s + 0.528·23-s − 5.59·27-s + 1.60·29-s − 3.79·31-s + 1.52·33-s + 8.92·37-s − 9.26·39-s + 1.86·41-s + 0.866·43-s + 1.19·47-s − 5.05·49-s − 4.98·51-s − 10.7·53-s − 1.52·57-s − 13.1·59-s − 3.12·61-s − 0.924·63-s − 2.26·67-s + 0.808·69-s + ⋯ |
L(s) = 1 | + 0.882·3-s + 0.527·7-s − 0.220·9-s + 0.301·11-s − 1.68·13-s − 0.791·17-s − 0.229·19-s + 0.465·21-s + 0.110·23-s − 1.07·27-s + 0.297·29-s − 0.680·31-s + 0.266·33-s + 1.46·37-s − 1.48·39-s + 0.291·41-s + 0.132·43-s + 0.173·47-s − 0.721·49-s − 0.698·51-s − 1.47·53-s − 0.202·57-s − 1.71·59-s − 0.400·61-s − 0.116·63-s − 0.276·67-s + 0.0973·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - 1.52T + 3T^{2} \) |
| 7 | \( 1 - 1.39T + 7T^{2} \) |
| 13 | \( 1 + 6.05T + 13T^{2} \) |
| 17 | \( 1 + 3.26T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 0.528T + 23T^{2} \) |
| 29 | \( 1 - 1.60T + 29T^{2} \) |
| 31 | \( 1 + 3.79T + 31T^{2} \) |
| 37 | \( 1 - 8.92T + 37T^{2} \) |
| 41 | \( 1 - 1.86T + 41T^{2} \) |
| 43 | \( 1 - 0.866T + 43T^{2} \) |
| 47 | \( 1 - 1.19T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 + 3.12T + 61T^{2} \) |
| 67 | \( 1 + 2.26T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 - 2.58T + 79T^{2} \) |
| 83 | \( 1 - 0.128T + 83T^{2} \) |
| 89 | \( 1 - 4.12T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.77749848156350532630341386479, −7.66328718696255154077118126343, −6.62340532059174174283073041674, −5.80840134854033487356741103098, −4.78725949116509166752919833130, −4.33239291914919662305595248509, −3.14849243987786485995698908944, −2.50810951274643824668381144977, −1.68214509556872672569426857033, 0,
1.68214509556872672569426857033, 2.50810951274643824668381144977, 3.14849243987786485995698908944, 4.33239291914919662305595248509, 4.78725949116509166752919833130, 5.80840134854033487356741103098, 6.62340532059174174283073041674, 7.66328718696255154077118126343, 7.77749848156350532630341386479