| L(s) = 1 | − 3-s + 7-s + 9-s − 4·11-s − 2·13-s − 2·17-s + 2·19-s − 21-s − 6·23-s − 27-s + 6·29-s − 6·31-s + 4·33-s + 4·37-s + 2·39-s − 4·43-s + 4·47-s + 49-s + 2·51-s + 2·53-s − 2·57-s − 4·59-s − 2·61-s + 63-s − 12·67-s + 6·69-s + 8·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.458·19-s − 0.218·21-s − 1.25·23-s − 0.192·27-s + 1.11·29-s − 1.07·31-s + 0.696·33-s + 0.657·37-s + 0.320·39-s − 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.274·53-s − 0.264·57-s − 0.520·59-s − 0.256·61-s + 0.125·63-s − 1.46·67-s + 0.722·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.050898657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.050898657\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.56279277731945914272912400131, −7.35048406859826782523602071262, −6.23306806472184064493666247970, −5.78489214933156331783686829675, −4.89697965516847316969092004032, −4.57114548767155620108541318156, −3.50101908861801476802011770116, −2.55181171513341671521163672538, −1.79768067173493383952295287520, −0.50273118136580517687805037565,
0.50273118136580517687805037565, 1.79768067173493383952295287520, 2.55181171513341671521163672538, 3.50101908861801476802011770116, 4.57114548767155620108541318156, 4.89697965516847316969092004032, 5.78489214933156331783686829675, 6.23306806472184064493666247970, 7.35048406859826782523602071262, 7.56279277731945914272912400131
