| L(s) = 1 | − 2·5-s − 7-s + 2·13-s − 6·17-s + 4·23-s − 25-s + 6·29-s + 8·31-s + 2·35-s + 10·37-s − 10·41-s + 12·43-s + 8·47-s + 49-s + 6·53-s + 4·59-s − 10·61-s − 4·65-s − 12·67-s + 4·71-s + 2·73-s − 8·79-s − 4·83-s + 12·85-s + 6·89-s − 2·91-s − 10·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 0.554·13-s − 1.45·17-s + 0.834·23-s − 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.338·35-s + 1.64·37-s − 1.56·41-s + 1.82·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.520·59-s − 1.28·61-s − 0.496·65-s − 1.46·67-s + 0.474·71-s + 0.234·73-s − 0.900·79-s − 0.439·83-s + 1.30·85-s + 0.635·89-s − 0.209·91-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 181944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 181944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.48608764363876, −13.00682458790985, −12.28479329397897, −12.01372891436254, −11.55143356781068, −10.99425166089655, −10.69229369823996, −10.16118156128901, −9.522578133077433, −9.028718845928797, −8.618910786005288, −8.159324096367988, −7.664181861919487, −7.047986733226919, −6.692013668753742, −6.123268651984801, −5.701001252407529, −4.828344244506092, −4.361073644833774, −4.109865126215108, −3.383730157724772, −2.685441860791918, −2.446377367587573, −1.329884092214256, −0.7792408339150788, 0,
0.7792408339150788, 1.329884092214256, 2.446377367587573, 2.685441860791918, 3.383730157724772, 4.109865126215108, 4.361073644833774, 4.828344244506092, 5.701001252407529, 6.123268651984801, 6.692013668753742, 7.047986733226919, 7.664181861919487, 8.159324096367988, 8.618910786005288, 9.028718845928797, 9.522578133077433, 10.16118156128901, 10.69229369823996, 10.99425166089655, 11.55143356781068, 12.01372891436254, 12.28479329397897, 13.00682458790985, 13.48608764363876
