| L(s) = 1 | − 2·4-s + 3·5-s − 4·13-s + 4·16-s + 6·17-s + 2·19-s − 6·20-s − 3·23-s + 4·25-s − 6·29-s − 5·31-s + 11·37-s − 6·41-s − 8·43-s + 8·52-s + 6·53-s − 9·59-s − 10·61-s − 8·64-s − 12·65-s + 5·67-s − 12·68-s − 9·71-s + 2·73-s − 4·76-s + 10·79-s + 12·80-s + ⋯ |
| L(s) = 1 | − 4-s + 1.34·5-s − 1.10·13-s + 16-s + 1.45·17-s + 0.458·19-s − 1.34·20-s − 0.625·23-s + 4/5·25-s − 1.11·29-s − 0.898·31-s + 1.80·37-s − 0.937·41-s − 1.21·43-s + 1.10·52-s + 0.824·53-s − 1.17·59-s − 1.28·61-s − 64-s − 1.48·65-s + 0.610·67-s − 1.45·68-s − 1.06·71-s + 0.234·73-s − 0.458·76-s + 1.12·79-s + 1.34·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.839130331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.839130331\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.37923460331289, −13.89314959937574, −13.51799881314263, −12.97305607074112, −12.57087476860696, −11.99886280740357, −11.48799214040589, −10.57123299539607, −10.13569879528575, −9.729979994154978, −9.409470216488211, −8.971474653656877, −8.079401890558164, −7.734534604415497, −7.148474753979967, −6.299632511561666, −5.650738711177625, −5.478409991967942, −4.845371236441877, −4.200397607383899, −3.400867625920917, −2.886871455403166, −1.951809902570155, −1.456339988495990, −0.4792524092538171,
0.4792524092538171, 1.456339988495990, 1.951809902570155, 2.886871455403166, 3.400867625920917, 4.200397607383899, 4.845371236441877, 5.478409991967942, 5.650738711177625, 6.299632511561666, 7.148474753979967, 7.734534604415497, 8.079401890558164, 8.971474653656877, 9.409470216488211, 9.729979994154978, 10.13569879528575, 10.57123299539607, 11.48799214040589, 11.99886280740357, 12.57087476860696, 12.97305607074112, 13.51799881314263, 13.89314959937574, 14.37923460331289
