| L(s) = 1 | − 4·5-s − 2·7-s + 3·11-s − 4·17-s + 2·19-s + 23-s + 11·25-s − 2·29-s − 8·31-s + 8·35-s − 2·37-s − 8·41-s + 4·43-s + 3·47-s − 3·49-s − 6·53-s − 12·55-s + 9·59-s − 14·61-s − 4·67-s + 11·73-s − 6·77-s − 10·79-s − 15·83-s + 16·85-s + 14·89-s − 8·95-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.755·7-s + 0.904·11-s − 0.970·17-s + 0.458·19-s + 0.208·23-s + 11/5·25-s − 0.371·29-s − 1.43·31-s + 1.35·35-s − 0.328·37-s − 1.24·41-s + 0.609·43-s + 0.437·47-s − 3/7·49-s − 0.824·53-s − 1.61·55-s + 1.17·59-s − 1.79·61-s − 0.488·67-s + 1.28·73-s − 0.683·77-s − 1.12·79-s − 1.64·83-s + 1.73·85-s + 1.48·89-s − 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3388736974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3388736974\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−14.18742424788642, −13.54486950823895, −12.96178205870769, −12.52637071475483, −12.10629048538414, −11.54677708944368, −11.21517752961282, −10.76569536446033, −10.09523298118109, −9.340075657339169, −9.024051395988277, −8.535422097708299, −7.894009261071386, −7.360994808892502, −6.921773113518987, −6.537855073253853, −5.774588554305537, −5.043760683841435, −4.387535805873438, −3.968895851511897, −3.382343709429086, −3.053549430124284, −2.012944035994114, −1.169572569110706, −0.2098714936749561,
0.2098714936749561, 1.169572569110706, 2.012944035994114, 3.053549430124284, 3.382343709429086, 3.968895851511897, 4.387535805873438, 5.043760683841435, 5.774588554305537, 6.537855073253853, 6.921773113518987, 7.360994808892502, 7.894009261071386, 8.535422097708299, 9.024051395988277, 9.340075657339169, 10.09523298118109, 10.76569536446033, 11.21517752961282, 11.54677708944368, 12.10629048538414, 12.52637071475483, 12.96178205870769, 13.54486950823895, 14.18742424788642
