| L(s) = 1 | − 2·5-s − 7-s − 4·13-s + 4·19-s − 4·23-s − 25-s + 10·29-s + 2·31-s + 2·35-s + 10·37-s − 4·43-s + 2·47-s + 49-s − 2·53-s − 6·59-s + 8·65-s − 8·67-s − 12·71-s + 12·73-s − 16·79-s − 4·83-s + 6·89-s + 4·91-s − 8·95-s − 10·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 1.10·13-s + 0.917·19-s − 0.834·23-s − 1/5·25-s + 1.85·29-s + 0.359·31-s + 0.338·35-s + 1.64·37-s − 0.609·43-s + 0.291·47-s + 1/7·49-s − 0.274·53-s − 0.781·59-s + 0.992·65-s − 0.977·67-s − 1.42·71-s + 1.40·73-s − 1.80·79-s − 0.439·83-s + 0.635·89-s + 0.419·91-s − 0.820·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.57611779825218, −14.02066696791545, −13.62085825922501, −12.99451554145046, −12.29070922899755, −12.08583392942874, −11.65514536130795, −11.09949968716310, −10.38779751571107, −9.831512161508484, −9.659188267046544, −8.809235690041264, −8.284426288762944, −7.667801475264110, −7.468982865171288, −6.705873856311600, −6.189825244150290, −5.559359353826264, −4.790987258726228, −4.389797045136993, −3.819789809407822, −2.873168262368571, −2.780151762136329, −1.673122663708715, −0.7755741783368163, 0,
0.7755741783368163, 1.673122663708715, 2.780151762136329, 2.873168262368571, 3.819789809407822, 4.389797045136993, 4.790987258726228, 5.559359353826264, 6.189825244150290, 6.705873856311600, 7.468982865171288, 7.667801475264110, 8.284426288762944, 8.809235690041264, 9.659188267046544, 9.831512161508484, 10.38779751571107, 11.09949968716310, 11.65514536130795, 12.08583392942874, 12.29070922899755, 12.99451554145046, 13.62085825922501, 14.02066696791545, 14.57611779825218
