L(s) = 1 | + 9·3-s + 26·7-s + 54·9-s + 59·11-s − 28·13-s − 5·17-s − 109·19-s + 234·21-s − 194·23-s + 243·27-s − 32·29-s − 10·31-s + 531·33-s + 198·37-s − 252·39-s + 117·41-s + 388·43-s − 68·47-s + 333·49-s − 45·51-s + 18·53-s − 981·57-s − 392·59-s − 710·61-s + 1.40e3·63-s − 253·67-s − 1.74e3·69-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 1.40·7-s + 2·9-s + 1.61·11-s − 0.597·13-s − 0.0713·17-s − 1.31·19-s + 2.43·21-s − 1.75·23-s + 1.73·27-s − 0.204·29-s − 0.0579·31-s + 2.80·33-s + 0.879·37-s − 1.03·39-s + 0.445·41-s + 1.37·43-s − 0.211·47-s + 0.970·49-s − 0.123·51-s + 0.0466·53-s − 2.27·57-s − 0.864·59-s − 1.49·61-s + 2.80·63-s − 0.461·67-s − 3.04·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.217150837\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.217150837\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 7 | \( 1 - 26 T + p^{3} T^{2} \) |
| 11 | \( 1 - 59 T + p^{3} T^{2} \) |
| 13 | \( 1 + 28 T + p^{3} T^{2} \) |
| 17 | \( 1 + 5 T + p^{3} T^{2} \) |
| 19 | \( 1 + 109 T + p^{3} T^{2} \) |
| 23 | \( 1 + 194 T + p^{3} T^{2} \) |
| 29 | \( 1 + 32 T + p^{3} T^{2} \) |
| 31 | \( 1 + 10 T + p^{3} T^{2} \) |
| 37 | \( 1 - 198 T + p^{3} T^{2} \) |
| 41 | \( 1 - 117 T + p^{3} T^{2} \) |
| 43 | \( 1 - 388 T + p^{3} T^{2} \) |
| 47 | \( 1 + 68 T + p^{3} T^{2} \) |
| 53 | \( 1 - 18 T + p^{3} T^{2} \) |
| 59 | \( 1 + 392 T + p^{3} T^{2} \) |
| 61 | \( 1 + 710 T + p^{3} T^{2} \) |
| 67 | \( 1 + 253 T + p^{3} T^{2} \) |
| 71 | \( 1 - 612 T + p^{3} T^{2} \) |
| 73 | \( 1 - 549 T + p^{3} T^{2} \) |
| 79 | \( 1 + 414 T + p^{3} T^{2} \) |
| 83 | \( 1 + 121 T + p^{3} T^{2} \) |
| 89 | \( 1 + 81 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1502 T + p^{3} T^{2} \) |
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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
−10.76189716094854609410180026356, −9.608495014587047179121376136936, −8.943123469000475603149527714086, −8.127931284476167627988067387117, −7.52746210854793021305012366627, −6.26112684748082050969754083726, −4.46677087712084593790776008492, −3.92946463868303334724598949814, −2.35916375917987048982854747755, −1.55445623047435528645373196435,
1.55445623047435528645373196435, 2.35916375917987048982854747755, 3.92946463868303334724598949814, 4.46677087712084593790776008492, 6.26112684748082050969754083726, 7.52746210854793021305012366627, 8.127931284476167627988067387117, 8.943123469000475603149527714086, 9.608495014587047179121376136936, 10.76189716094854609410180026356