L(s) = 1 | + 3-s + 7-s + 9-s − 5·11-s − 2·13-s + 2·17-s + 19-s + 21-s − 23-s + 27-s − 4·29-s − 5·33-s + 6·37-s − 2·39-s − 3·41-s − 2·43-s − 13·47-s + 49-s + 2·51-s + 3·53-s + 57-s + 4·59-s − 10·61-s + 63-s + 14·67-s − 69-s − 14·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.554·13-s + 0.485·17-s + 0.229·19-s + 0.218·21-s − 0.208·23-s + 0.192·27-s − 0.742·29-s − 0.870·33-s + 0.986·37-s − 0.320·39-s − 0.468·41-s − 0.304·43-s − 1.89·47-s + 1/7·49-s + 0.280·51-s + 0.412·53-s + 0.132·57-s + 0.520·59-s − 1.28·61-s + 0.125·63-s + 1.71·67-s − 0.120·69-s − 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.611882872\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.611882872\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−13.11973617876575, −12.69651726430484, −12.28632243443163, −11.63428025903160, −11.22195481531518, −10.71911703328967, −10.18687889110864, −9.724606221953718, −9.512503387189424, −8.662969730051289, −8.280130857882182, −7.815901400051948, −7.551379273601655, −6.949336950934670, −6.381537203333721, −5.576031094498715, −5.328045963726301, −4.753748959121431, −4.232246198816240, −3.528361083141330, −2.970885912420723, −2.534321222853600, −1.913620565978435, −1.305917676347670, −0.3356606613659696,
0.3356606613659696, 1.305917676347670, 1.913620565978435, 2.534321222853600, 2.970885912420723, 3.528361083141330, 4.232246198816240, 4.753748959121431, 5.328045963726301, 5.576031094498715, 6.381537203333721, 6.949336950934670, 7.551379273601655, 7.815901400051948, 8.280130857882182, 8.662969730051289, 9.512503387189424, 9.724606221953718, 10.18687889110864, 10.71911703328967, 11.22195481531518, 11.63428025903160, 12.28632243443163, 12.69651726430484, 13.11973617876575