| L(s) = 1 | + 3-s − 2·5-s + 9-s − 4·11-s + 2·13-s − 2·15-s − 2·17-s + 4·19-s − 8·23-s − 25-s + 27-s − 6·29-s + 8·31-s − 4·33-s + 6·37-s + 2·39-s + 4·43-s − 2·45-s − 7·49-s − 2·51-s + 2·53-s + 8·55-s + 4·57-s + 4·59-s − 2·61-s − 4·65-s + 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.516·15-s − 0.485·17-s + 0.917·19-s − 1.66·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.696·33-s + 0.986·37-s + 0.320·39-s + 0.609·43-s − 0.298·45-s − 49-s − 0.280·51-s + 0.274·53-s + 1.07·55-s + 0.529·57-s + 0.520·59-s − 0.256·61-s − 0.496·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 41 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.24109284359181, −14.45482549658490, −14.01057469426522, −13.42826463376771, −13.10312115714026, −12.46293288520022, −11.84669386403058, −11.42774934145330, −10.92399802593475, −10.18771134411198, −9.828685058119624, −9.200141813345848, −8.459779177782159, −8.026186869701344, −7.698711958829984, −7.221468198494497, −6.277900658041318, −5.873746920015955, −5.027359618217080, −4.452073068397599, −3.802822922520274, −3.347597208893214, −2.531350183300979, −1.994385323129170, −0.8996167545932216, 0,
0.8996167545932216, 1.994385323129170, 2.531350183300979, 3.347597208893214, 3.802822922520274, 4.452073068397599, 5.027359618217080, 5.873746920015955, 6.277900658041318, 7.221468198494497, 7.698711958829984, 8.026186869701344, 8.459779177782159, 9.200141813345848, 9.828685058119624, 10.18771134411198, 10.92399802593475, 11.42774934145330, 11.84669386403058, 12.46293288520022, 13.10312115714026, 13.42826463376771, 14.01057469426522, 14.45482549658490, 15.24109284359181
