L(s) = 1 | − 2-s − 3·3-s − 4-s − 4·5-s + 3·6-s + 7-s + 3·8-s + 6·9-s + 4·10-s + 3·12-s − 2·13-s − 14-s + 12·15-s − 16-s − 6·18-s − 7·19-s + 4·20-s − 3·21-s − 4·23-s − 9·24-s + 11·25-s + 2·26-s − 9·27-s − 28-s − 3·29-s − 12·30-s − 7·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.73·3-s − 1/2·4-s − 1.78·5-s + 1.22·6-s + 0.377·7-s + 1.06·8-s + 2·9-s + 1.26·10-s + 0.866·12-s − 0.554·13-s − 0.267·14-s + 3.09·15-s − 1/4·16-s − 1.41·18-s − 1.60·19-s + 0.894·20-s − 0.654·21-s − 0.834·23-s − 1.83·24-s + 11/5·25-s + 0.392·26-s − 1.73·27-s − 0.188·28-s − 0.557·29-s − 2.19·30-s − 1.25·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−8.325324976074901463143938700682, −7.49270934066170001209592011836, −7.08703260812682809283556063921, −5.96229341033422925854318134461, −4.93740487759233365545594500230, −4.43464217605583627658799219424, −3.74899416217715636480760090572, −1.59186266120372282824621797255, 0, 0,
1.59186266120372282824621797255, 3.74899416217715636480760090572, 4.43464217605583627658799219424, 4.93740487759233365545594500230, 5.96229341033422925854318134461, 7.08703260812682809283556063921, 7.49270934066170001209592011836, 8.325324976074901463143938700682