| L(s) = 1 | − 3-s + 9-s − 4·11-s − 2·13-s − 6·17-s + 4·19-s − 27-s − 2·29-s + 4·33-s − 6·37-s + 2·39-s − 2·41-s − 4·43-s + 6·51-s − 6·53-s − 4·57-s + 12·59-s + 2·61-s + 4·67-s − 6·73-s + 16·79-s + 81-s + 12·83-s + 2·87-s + 14·89-s + 18·97-s − 4·99-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 1.45·17-s + 0.917·19-s − 0.192·27-s − 0.371·29-s + 0.696·33-s − 0.986·37-s + 0.320·39-s − 0.312·41-s − 0.609·43-s + 0.840·51-s − 0.824·53-s − 0.529·57-s + 1.56·59-s + 0.256·61-s + 0.488·67-s − 0.702·73-s + 1.80·79-s + 1/9·81-s + 1.31·83-s + 0.214·87-s + 1.48·89-s + 1.82·97-s − 0.402·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.74623105342564, −13.94210505574186, −13.42464892964383, −13.17062254204669, −12.55627864429200, −12.00266064646939, −11.57151712227008, −10.97881829685371, −10.57412271993644, −10.08237968176316, −9.494016013502582, −9.001150019611454, −8.279947722184537, −7.817758216537041, −7.190754873331540, −6.753656466805761, −6.177516238747682, −5.333280777617961, −5.131052729289558, −4.567199216758859, −3.759499759032620, −3.118905637749644, −2.312975590812199, −1.855832941483494, −0.7205351204892218, 0,
0.7205351204892218, 1.855832941483494, 2.312975590812199, 3.118905637749644, 3.759499759032620, 4.567199216758859, 5.131052729289558, 5.333280777617961, 6.177516238747682, 6.753656466805761, 7.190754873331540, 7.817758216537041, 8.279947722184537, 9.001150019611454, 9.494016013502582, 10.08237968176316, 10.57412271993644, 10.97881829685371, 11.57151712227008, 12.00266064646939, 12.55627864429200, 13.17062254204669, 13.42464892964383, 13.94210505574186, 14.74623105342564
