| L(s) = 1 | + 3-s + 3·7-s + 9-s + 11-s + 13-s + 5·17-s + 8·19-s + 3·21-s + 27-s − 29-s + 3·31-s + 33-s − 8·37-s + 39-s − 2·41-s + 8·43-s + 11·47-s + 2·49-s + 5·51-s − 11·53-s + 8·57-s − 5·59-s − 61-s + 3·63-s + 3·67-s + 16·71-s + 4·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 1.21·17-s + 1.83·19-s + 0.654·21-s + 0.192·27-s − 0.185·29-s + 0.538·31-s + 0.174·33-s − 1.31·37-s + 0.160·39-s − 0.312·41-s + 1.21·43-s + 1.60·47-s + 2/7·49-s + 0.700·51-s − 1.51·53-s + 1.05·57-s − 0.650·59-s − 0.128·61-s + 0.377·63-s + 0.366·67-s + 1.89·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.216631675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.216631675\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.27314672229864, −13.85782505707645, −13.58695369963909, −12.58903606171932, −12.24508511398831, −11.86147308726925, −11.07048023610828, −10.93216030604195, −9.992080847377608, −9.743223667036362, −9.063640547266536, −8.619218827240478, −7.958458276859074, −7.588984393286508, −7.255526321557707, −6.370554569130077, −5.769474000877986, −5.045837209921616, −4.870205863201469, −3.818798413124243, −3.517853170771577, −2.797611118498165, −2.015471007116494, −1.310012406930663, −0.8430286750635895,
0.8430286750635895, 1.310012406930663, 2.015471007116494, 2.797611118498165, 3.517853170771577, 3.818798413124243, 4.870205863201469, 5.045837209921616, 5.769474000877986, 6.370554569130077, 7.255526321557707, 7.588984393286508, 7.958458276859074, 8.619218827240478, 9.063640547266536, 9.743223667036362, 9.992080847377608, 10.93216030604195, 11.07048023610828, 11.86147308726925, 12.24508511398831, 12.58903606171932, 13.58695369963909, 13.85782505707645, 14.27314672229864
