| L(s) = 1 | + 2·2-s + 2·4-s + 2·7-s + 4·14-s − 4·16-s + 2·17-s − 4·19-s + 23-s + 4·28-s − 5·29-s + 10·31-s − 8·32-s + 4·34-s + 2·37-s − 8·38-s − 10·41-s − 11·43-s + 2·46-s + 8·47-s − 3·49-s + 9·53-s − 10·58-s + 6·59-s + 7·61-s + 20·62-s − 8·64-s − 12·67-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.755·7-s + 1.06·14-s − 16-s + 0.485·17-s − 0.917·19-s + 0.208·23-s + 0.755·28-s − 0.928·29-s + 1.79·31-s − 1.41·32-s + 0.685·34-s + 0.328·37-s − 1.29·38-s − 1.56·41-s − 1.67·43-s + 0.294·46-s + 1.16·47-s − 3/7·49-s + 1.23·53-s − 1.31·58-s + 0.781·59-s + 0.896·61-s + 2.54·62-s − 64-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.129271746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.129271746\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.77588239339347, −14.31371189558869, −13.78133853240876, −13.30153816266529, −12.95436049467673, −12.21256235428447, −11.83787990194432, −11.43822496059069, −10.84174353897841, −10.16087141379383, −9.712834919666334, −8.732039674300445, −8.482159703462754, −7.799805728277381, −7.015970039194783, −6.572797242529137, −5.941279294178502, −5.319910582065082, −4.879491707832299, −4.330588022031889, −3.709783057114707, −3.122540042801836, −2.324745196168206, −1.755665103441992, −0.6428110167454213,
0.6428110167454213, 1.755665103441992, 2.324745196168206, 3.122540042801836, 3.709783057114707, 4.330588022031889, 4.879491707832299, 5.319910582065082, 5.941279294178502, 6.572797242529137, 7.015970039194783, 7.799805728277381, 8.482159703462754, 8.732039674300445, 9.712834919666334, 10.16087141379383, 10.84174353897841, 11.43822496059069, 11.83787990194432, 12.21256235428447, 12.95436049467673, 13.30153816266529, 13.78133853240876, 14.31371189558869, 14.77588239339347
