| L(s) = 1 | + 3-s − 2·7-s + 9-s − 2·11-s − 13-s + 2·17-s − 2·19-s − 2·21-s − 4·23-s + 27-s − 2·29-s + 2·31-s − 2·33-s − 6·37-s − 39-s − 2·41-s + 8·43-s − 6·47-s − 3·49-s + 2·51-s + 6·53-s − 2·57-s + 6·59-s + 2·61-s − 2·63-s + 14·67-s − 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.485·17-s − 0.458·19-s − 0.436·21-s − 0.834·23-s + 0.192·27-s − 0.371·29-s + 0.359·31-s − 0.348·33-s − 0.986·37-s − 0.160·39-s − 0.312·41-s + 1.21·43-s − 0.875·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s − 0.264·57-s + 0.781·59-s + 0.256·61-s − 0.251·63-s + 1.71·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.45868580756344, −14.10431831826557, −13.42532397988996, −13.06778046796709, −12.58428169163090, −12.13802792897204, −11.53456109128709, −10.89266062145704, −10.18696348526201, −10.06247079772347, −9.406267764167175, −8.919200924980222, −8.228278576644784, −7.917432447260750, −7.269681418701082, −6.692923231292101, −6.222479716005011, −5.459558257350762, −5.035213990573919, −4.182021240029237, −3.662253125450257, −3.135261063926899, −2.381309581950162, −1.956807668079155, −0.8755055653352857, 0,
0.8755055653352857, 1.956807668079155, 2.381309581950162, 3.135261063926899, 3.662253125450257, 4.182021240029237, 5.035213990573919, 5.459558257350762, 6.222479716005011, 6.692923231292101, 7.269681418701082, 7.917432447260750, 8.228278576644784, 8.919200924980222, 9.406267764167175, 10.06247079772347, 10.18696348526201, 10.89266062145704, 11.53456109128709, 12.13802792897204, 12.58428169163090, 13.06778046796709, 13.42532397988996, 14.10431831826557, 14.45868580756344
