L(s) = 1 | − 2.49·3-s + 2.92·7-s + 3.24·9-s + 4.10·11-s + 0.0122·13-s − 0.155·17-s − 4.32·19-s − 7.31·21-s − 23-s − 0.616·27-s − 6.79·29-s − 2.20·31-s − 10.2·33-s + 4.60·37-s − 0.0307·39-s − 7.67·41-s + 8.38·43-s + 6.38·47-s + 1.57·49-s + 0.388·51-s + 7.80·53-s + 10.8·57-s + 14.1·59-s + 7.05·61-s + 9.50·63-s + 7.31·67-s + 2.49·69-s + ⋯ |
L(s) = 1 | − 1.44·3-s + 1.10·7-s + 1.08·9-s + 1.23·11-s + 0.00341·13-s − 0.0376·17-s − 0.992·19-s − 1.59·21-s − 0.208·23-s − 0.118·27-s − 1.26·29-s − 0.396·31-s − 1.78·33-s + 0.757·37-s − 0.00492·39-s − 1.19·41-s + 1.27·43-s + 0.930·47-s + 0.224·49-s + 0.0543·51-s + 1.07·53-s + 1.43·57-s + 1.83·59-s + 0.903·61-s + 1.19·63-s + 0.893·67-s + 0.300·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.376610027\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.376610027\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + 2.49T + 3T^{2} \) |
| 7 | \( 1 - 2.92T + 7T^{2} \) |
| 11 | \( 1 - 4.10T + 11T^{2} \) |
| 13 | \( 1 - 0.0122T + 13T^{2} \) |
| 17 | \( 1 + 0.155T + 17T^{2} \) |
| 19 | \( 1 + 4.32T + 19T^{2} \) |
| 29 | \( 1 + 6.79T + 29T^{2} \) |
| 31 | \( 1 + 2.20T + 31T^{2} \) |
| 37 | \( 1 - 4.60T + 37T^{2} \) |
| 41 | \( 1 + 7.67T + 41T^{2} \) |
| 43 | \( 1 - 8.38T + 43T^{2} \) |
| 47 | \( 1 - 6.38T + 47T^{2} \) |
| 53 | \( 1 - 7.80T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 7.05T + 61T^{2} \) |
| 67 | \( 1 - 7.31T + 67T^{2} \) |
| 71 | \( 1 - 5.84T + 71T^{2} \) |
| 73 | \( 1 + 0.727T + 73T^{2} \) |
| 79 | \( 1 + 4.81T + 79T^{2} \) |
| 83 | \( 1 - 7.75T + 83T^{2} \) |
| 89 | \( 1 - 6.77T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−7.58475444838598001840032212817, −6.82129574378435344914777534267, −6.38418529067488577203167062782, −5.49839074035895375605116251658, −5.22859854268539532399364410990, −4.11811321837665273550304357858, −3.98749549536582014899176355525, −2.35384921319855394836787398193, −1.52487961210301294606591150966, −0.64959041423610141885701632661,
0.64959041423610141885701632661, 1.52487961210301294606591150966, 2.35384921319855394836787398193, 3.98749549536582014899176355525, 4.11811321837665273550304357858, 5.22859854268539532399364410990, 5.49839074035895375605116251658, 6.38418529067488577203167062782, 6.82129574378435344914777534267, 7.58475444838598001840032212817