| L(s) = 1 | − 3-s + 2·7-s + 9-s − 11-s − 4·13-s + 6·17-s − 4·19-s − 2·21-s + 6·23-s − 27-s − 6·29-s − 8·31-s + 33-s − 10·37-s + 4·39-s + 6·41-s − 8·43-s − 6·47-s − 3·49-s − 6·51-s + 4·57-s − 8·61-s + 2·63-s + 4·67-s − 6·69-s − 6·71-s − 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 1.45·17-s − 0.917·19-s − 0.436·21-s + 1.25·23-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.174·33-s − 1.64·37-s + 0.640·39-s + 0.937·41-s − 1.21·43-s − 0.875·47-s − 3/7·49-s − 0.840·51-s + 0.529·57-s − 1.02·61-s + 0.251·63-s + 0.488·67-s − 0.722·69-s − 0.712·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9888569856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9888569856\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.48750570849119, −14.16981722730779, −13.21498817800304, −12.92514180328683, −12.35062360516674, −11.97214999297315, −11.32083682423414, −10.89389415331284, −10.46208258093391, −9.843059835897315, −9.359036739878801, −8.728194060745225, −8.071816477444213, −7.560560133310865, −7.122694022112454, −6.590443226616901, −5.617348367798935, −5.392678945483431, −4.875615062893466, −4.256399314858698, −3.444246388819000, −2.878054799049608, −1.842770383814984, −1.522473234852986, −0.3517732063552715,
0.3517732063552715, 1.522473234852986, 1.842770383814984, 2.878054799049608, 3.444246388819000, 4.256399314858698, 4.875615062893466, 5.392678945483431, 5.617348367798935, 6.590443226616901, 7.122694022112454, 7.560560133310865, 8.071816477444213, 8.728194060745225, 9.359036739878801, 9.843059835897315, 10.46208258093391, 10.89389415331284, 11.32083682423414, 11.97214999297315, 12.35062360516674, 12.92514180328683, 13.21498817800304, 14.16981722730779, 14.48750570849119
