| L(s) = 1 | − 3-s + 2·7-s + 9-s + 4·11-s + 6·13-s − 2·17-s − 2·21-s + 6·23-s − 27-s − 2·29-s − 4·31-s − 4·33-s − 6·39-s + 6·41-s + 43-s − 10·47-s − 3·49-s + 2·51-s + 4·59-s − 12·61-s + 2·63-s − 4·67-s − 6·69-s + 2·73-s + 8·77-s − 16·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.485·17-s − 0.436·21-s + 1.25·23-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.696·33-s − 0.960·39-s + 0.937·41-s + 0.152·43-s − 1.45·47-s − 3/7·49-s + 0.280·51-s + 0.520·59-s − 1.53·61-s + 0.251·63-s − 0.488·67-s − 0.722·69-s + 0.234·73-s + 0.911·77-s − 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.711006447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.711006447\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.62182633447071, −14.00104920158507, −13.37731922749439, −12.98078358523481, −12.47906961466144, −11.69073531537497, −11.30183475091974, −11.10869223468564, −10.58844676358428, −9.807081519617501, −9.154315109347415, −8.821389337320196, −8.295630881869830, −7.574713106329782, −6.977661187725407, −6.472834714770594, −5.941711100252261, −5.442179305240573, −4.619558272573445, −4.261443792505407, −3.563175713695737, −2.950070606162943, −1.687726278115102, −1.490397988325553, −0.6362243370798666,
0.6362243370798666, 1.490397988325553, 1.687726278115102, 2.950070606162943, 3.563175713695737, 4.261443792505407, 4.619558272573445, 5.442179305240573, 5.941711100252261, 6.472834714770594, 6.977661187725407, 7.574713106329782, 8.295630881869830, 8.821389337320196, 9.154315109347415, 9.807081519617501, 10.58844676358428, 11.10869223468564, 11.30183475091974, 11.69073531537497, 12.47906961466144, 12.98078358523481, 13.37731922749439, 14.00104920158507, 14.62182633447071
