L(s) = 1 | + 3-s + 2·7-s + 9-s + 13-s + 2·17-s − 2·19-s + 2·21-s + 8·23-s − 5·25-s + 27-s + 6·29-s + 2·31-s − 6·37-s + 39-s + 4·43-s + 8·47-s − 3·49-s + 2·51-s − 6·53-s − 2·57-s + 4·59-s + 2·61-s + 2·63-s + 2·67-s + 8·69-s + 4·71-s − 2·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.277·13-s + 0.485·17-s − 0.458·19-s + 0.436·21-s + 1.66·23-s − 25-s + 0.192·27-s + 1.11·29-s + 0.359·31-s − 0.986·37-s + 0.160·39-s + 0.609·43-s + 1.16·47-s − 3/7·49-s + 0.280·51-s − 0.824·53-s − 0.264·57-s + 0.520·59-s + 0.256·61-s + 0.251·63-s + 0.244·67-s + 0.963·69-s + 0.474·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.308753302\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.308753302\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−9.614324717724064020306346133933, −8.772746136966357733511362932357, −8.170210140292499884928170378646, −7.37444340174235339103368042805, −6.48798103863965861426704983959, −5.36708441859909895042575363132, −4.53138639637399252599647393052, −3.51100966248523097859158706461, −2.43766757243998842407464205467, −1.21909654633704707619269809337,
1.21909654633704707619269809337, 2.43766757243998842407464205467, 3.51100966248523097859158706461, 4.53138639637399252599647393052, 5.36708441859909895042575363132, 6.48798103863965861426704983959, 7.37444340174235339103368042805, 8.170210140292499884928170378646, 8.772746136966357733511362932357, 9.614324717724064020306346133933