| L(s) = 1 | + 5-s − 4·7-s − 11-s + 17-s + 6·19-s + 2·23-s + 25-s − 5·31-s − 4·35-s − 2·37-s − 6·41-s + 5·43-s − 4·47-s + 9·49-s + 6·53-s − 55-s − 7·59-s − 8·61-s + 2·67-s − 8·71-s + 7·73-s + 4·77-s − 16·79-s − 9·83-s + 85-s − 9·89-s + 6·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s − 0.301·11-s + 0.242·17-s + 1.37·19-s + 0.417·23-s + 1/5·25-s − 0.898·31-s − 0.676·35-s − 0.328·37-s − 0.937·41-s + 0.762·43-s − 0.583·47-s + 9/7·49-s + 0.824·53-s − 0.134·55-s − 0.911·59-s − 1.02·61-s + 0.244·67-s − 0.949·71-s + 0.819·73-s + 0.455·77-s − 1.80·79-s − 0.987·83-s + 0.108·85-s − 0.953·89-s + 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9745173741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9745173741\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−12.72011871921228, −12.19047261215257, −11.75990200579074, −11.22549843765240, −10.65212276846633, −10.24970281236650, −9.800228104109315, −9.482060562523699, −9.049246744563681, −8.602583836311594, −7.931666084108679, −7.333520845533358, −7.068553834031077, −6.572247642076550, −5.953112491503835, −5.633595309473865, −5.190133071032250, −4.548743028413503, −3.866249558424314, −3.323550320345680, −2.983585387474163, −2.519324424591477, −1.650377019668067, −1.143905985782544, −0.2672541223228142,
0.2672541223228142, 1.143905985782544, 1.650377019668067, 2.519324424591477, 2.983585387474163, 3.323550320345680, 3.866249558424314, 4.548743028413503, 5.190133071032250, 5.633595309473865, 5.953112491503835, 6.572247642076550, 7.068553834031077, 7.333520845533358, 7.931666084108679, 8.602583836311594, 9.049246744563681, 9.482060562523699, 9.800228104109315, 10.24970281236650, 10.65212276846633, 11.22549843765240, 11.75990200579074, 12.19047261215257, 12.72011871921228
