| L(s) = 1 | + 3·3-s + 4·7-s + 6·9-s + 2·11-s + 5·13-s − 4·17-s − 2·19-s + 12·21-s − 23-s + 9·27-s − 7·29-s − 3·31-s + 6·33-s − 2·37-s + 15·39-s − 9·41-s + 8·43-s − 9·47-s + 9·49-s − 12·51-s − 2·53-s − 6·57-s − 2·61-s + 24·63-s − 14·67-s − 3·69-s − 3·71-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.51·7-s + 2·9-s + 0.603·11-s + 1.38·13-s − 0.970·17-s − 0.458·19-s + 2.61·21-s − 0.208·23-s + 1.73·27-s − 1.29·29-s − 0.538·31-s + 1.04·33-s − 0.328·37-s + 2.40·39-s − 1.40·41-s + 1.21·43-s − 1.31·47-s + 9/7·49-s − 1.68·51-s − 0.274·53-s − 0.794·57-s − 0.256·61-s + 3.02·63-s − 1.71·67-s − 0.361·69-s − 0.356·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.210147284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.210147284\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.792802746744538848579641860594, −8.417205547335714423363480484600, −7.72155011614594458693352981488, −6.97259534717922181480572966965, −5.92939816039003138214906528127, −4.68020147118361724754014300098, −4.02664742108510952615562824417, −3.27927865891199452005330795104, −1.97345603016417977923824888144, −1.56839486015603156767283193462,
1.56839486015603156767283193462, 1.97345603016417977923824888144, 3.27927865891199452005330795104, 4.02664742108510952615562824417, 4.68020147118361724754014300098, 5.92939816039003138214906528127, 6.97259534717922181480572966965, 7.72155011614594458693352981488, 8.417205547335714423363480484600, 8.792802746744538848579641860594
