| L(s) = 1 | − 4·2-s + 16·4-s + 22·7-s − 64·8-s + 768·11-s + 46·13-s − 88·14-s + 256·16-s + 378·17-s + 1.10e3·19-s − 3.07e3·22-s − 1.98e3·23-s − 184·26-s + 352·28-s + 5.61e3·29-s − 3.98e3·31-s − 1.02e3·32-s − 1.51e3·34-s + 142·37-s − 4.40e3·38-s − 1.54e3·41-s + 5.02e3·43-s + 1.22e4·44-s + 7.94e3·46-s + 2.47e4·47-s − 1.63e4·49-s + 736·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.169·7-s − 0.353·8-s + 1.91·11-s + 0.0754·13-s − 0.119·14-s + 1/4·16-s + 0.317·17-s + 0.699·19-s − 1.35·22-s − 0.782·23-s − 0.0533·26-s + 0.0848·28-s + 1.23·29-s − 0.745·31-s − 0.176·32-s − 0.224·34-s + 0.0170·37-s − 0.494·38-s − 0.143·41-s + 0.414·43-s + 0.956·44-s + 0.553·46-s + 1.63·47-s − 0.971·49-s + 0.0377·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.847750820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.847750820\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 22 T + p^{5} T^{2} \) |
| 11 | \( 1 - 768 T + p^{5} T^{2} \) |
| 13 | \( 1 - 46 T + p^{5} T^{2} \) |
| 17 | \( 1 - 378 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1100 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1986 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5610 T + p^{5} T^{2} \) |
| 31 | \( 1 + 3988 T + p^{5} T^{2} \) |
| 37 | \( 1 - 142 T + p^{5} T^{2} \) |
| 41 | \( 1 + 1542 T + p^{5} T^{2} \) |
| 43 | \( 1 - 5026 T + p^{5} T^{2} \) |
| 47 | \( 1 - 24738 T + p^{5} T^{2} \) |
| 53 | \( 1 + 14166 T + p^{5} T^{2} \) |
| 59 | \( 1 + 28380 T + p^{5} T^{2} \) |
| 61 | \( 1 - 5522 T + p^{5} T^{2} \) |
| 67 | \( 1 - 24742 T + p^{5} T^{2} \) |
| 71 | \( 1 + 42372 T + p^{5} T^{2} \) |
| 73 | \( 1 - 52126 T + p^{5} T^{2} \) |
| 79 | \( 1 + 39640 T + p^{5} T^{2} \) |
| 83 | \( 1 + 59826 T + p^{5} T^{2} \) |
| 89 | \( 1 + 57690 T + p^{5} T^{2} \) |
| 97 | \( 1 - 144382 T + p^{5} T^{2} \) |
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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
−10.11363356959375769322391803899, −9.345051154810690820620994215593, −8.627565195023354015991106012258, −7.59612181304230248695314577617, −6.67915953686457636835841873936, −5.82726575202111764838506502729, −4.36890336873441971984388428619, −3.28191578673104306357513045503, −1.77165232192124183688571663183, −0.827744371892319461989208705724,
0.827744371892319461989208705724, 1.77165232192124183688571663183, 3.28191578673104306357513045503, 4.36890336873441971984388428619, 5.82726575202111764838506502729, 6.67915953686457636835841873936, 7.59612181304230248695314577617, 8.627565195023354015991106012258, 9.345051154810690820620994215593, 10.11363356959375769322391803899
