| L(s) = 1 | + 2-s + 7·3-s − 7·4-s + 7·6-s + 6·7-s − 15·8-s + 22·9-s − 43·11-s − 49·12-s − 28·13-s + 6·14-s + 41·16-s + 91·17-s + 22·18-s − 35·19-s + 42·21-s − 43·22-s + 162·23-s − 105·24-s − 28·26-s − 35·27-s − 42·28-s + 160·29-s + 42·31-s + 161·32-s − 301·33-s + 91·34-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 1.34·3-s − 7/8·4-s + 0.476·6-s + 0.323·7-s − 0.662·8-s + 0.814·9-s − 1.17·11-s − 1.17·12-s − 0.597·13-s + 0.114·14-s + 0.640·16-s + 1.29·17-s + 0.288·18-s − 0.422·19-s + 0.436·21-s − 0.416·22-s + 1.46·23-s − 0.893·24-s − 0.211·26-s − 0.249·27-s − 0.283·28-s + 1.02·29-s + 0.243·31-s + 0.889·32-s − 1.58·33-s + 0.459·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.565244413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.565244413\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 7 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 43 T + p^{3} T^{2} \) |
| 13 | \( 1 + 28 T + p^{3} T^{2} \) |
| 17 | \( 1 - 91 T + p^{3} T^{2} \) |
| 19 | \( 1 + 35 T + p^{3} T^{2} \) |
| 23 | \( 1 - 162 T + p^{3} T^{2} \) |
| 29 | \( 1 - 160 T + p^{3} T^{2} \) |
| 31 | \( 1 - 42 T + p^{3} T^{2} \) |
| 37 | \( 1 + 314 T + p^{3} T^{2} \) |
| 41 | \( 1 + 203 T + p^{3} T^{2} \) |
| 43 | \( 1 - 92 T + p^{3} T^{2} \) |
| 47 | \( 1 - 196 T + p^{3} T^{2} \) |
| 53 | \( 1 - 82 T + p^{3} T^{2} \) |
| 59 | \( 1 + 280 T + p^{3} T^{2} \) |
| 61 | \( 1 + 518 T + p^{3} T^{2} \) |
| 67 | \( 1 - 141 T + p^{3} T^{2} \) |
| 71 | \( 1 - 412 T + p^{3} T^{2} \) |
| 73 | \( 1 + 763 T + p^{3} T^{2} \) |
| 79 | \( 1 - 510 T + p^{3} T^{2} \) |
| 83 | \( 1 - 777 T + p^{3} T^{2} \) |
| 89 | \( 1 + 945 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1246 T + p^{3} T^{2} \) |
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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
−17.21572709070106100516398473425, −15.34942168053280394849793785256, −14.45265091939715657978905574226, −13.57010387874695640408048454315, −12.45877293985707783572300704744, −10.18178389633868292998924791706, −8.842597663368459781858121888813, −7.78482906717815976923379874673, −5.03392178674914352750806245984, −3.09665238839614084696729405245,
3.09665238839614084696729405245, 5.03392178674914352750806245984, 7.78482906717815976923379874673, 8.842597663368459781858121888813, 10.18178389633868292998924791706, 12.45877293985707783572300704744, 13.57010387874695640408048454315, 14.45265091939715657978905574226, 15.34942168053280394849793785256, 17.21572709070106100516398473425
