L(s) = 1 | − 2.19·3-s + 0.992·7-s + 1.83·9-s − 2·11-s + 3.37·13-s − 2.89·17-s + 2.58·19-s − 2.18·21-s − 4.54·23-s + 2.56·27-s − 5.38·29-s − 0.136·31-s + 4.39·33-s + 2.14·37-s − 7.41·39-s + 8.63·41-s + 4.64·43-s + 9.92·47-s − 6.01·49-s + 6.36·51-s − 7.56·53-s − 5.68·57-s − 4.91·59-s − 2.76·61-s + 1.81·63-s + 2.18·67-s + 10.0·69-s + ⋯ |
L(s) = 1 | − 1.26·3-s + 0.375·7-s + 0.611·9-s − 0.603·11-s + 0.935·13-s − 0.702·17-s + 0.592·19-s − 0.476·21-s − 0.948·23-s + 0.493·27-s − 0.999·29-s − 0.0245·31-s + 0.765·33-s + 0.353·37-s − 1.18·39-s + 1.34·41-s + 0.708·43-s + 1.44·47-s − 0.859·49-s + 0.891·51-s − 1.03·53-s − 0.752·57-s − 0.640·59-s − 0.354·61-s + 0.229·63-s + 0.267·67-s + 1.20·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 3 | \( 1 + 2.19T + 3T^{2} \) |
| 7 | \( 1 - 0.992T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 3.37T + 13T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 19 | \( 1 - 2.58T + 19T^{2} \) |
| 23 | \( 1 + 4.54T + 23T^{2} \) |
| 29 | \( 1 + 5.38T + 29T^{2} \) |
| 31 | \( 1 + 0.136T + 31T^{2} \) |
| 37 | \( 1 - 2.14T + 37T^{2} \) |
| 41 | \( 1 - 8.63T + 41T^{2} \) |
| 43 | \( 1 - 4.64T + 43T^{2} \) |
| 47 | \( 1 - 9.92T + 47T^{2} \) |
| 53 | \( 1 + 7.56T + 53T^{2} \) |
| 59 | \( 1 + 4.91T + 59T^{2} \) |
| 61 | \( 1 + 2.76T + 61T^{2} \) |
| 67 | \( 1 - 2.18T + 67T^{2} \) |
| 71 | \( 1 + 9.64T + 71T^{2} \) |
| 73 | \( 1 + 0.775T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 + 1.77T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−7.40345703646152516364836806296, −6.31068688358118564863749470103, −5.99754704782864634296779526417, −5.40350459884153726334726088364, −4.62993794640346102083458171426, −4.04315216652912820874023310477, −3.01664469380271316554405019863, −2.01023058139772382159728385800, −1.02411419896622267627210846103, 0,
1.02411419896622267627210846103, 2.01023058139772382159728385800, 3.01664469380271316554405019863, 4.04315216652912820874023310477, 4.62993794640346102083458171426, 5.40350459884153726334726088364, 5.99754704782864634296779526417, 6.31068688358118564863749470103, 7.40345703646152516364836806296