| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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
−8.341462190423398317689978413119, −7.84357206540786593726080749054, −6.97191988730281640323478124432, −5.95482100145756852622378928374, −5.23101901361279341448717228242, −4.95037144105201661968564931705, −3.79045627784860642299428481789, −2.13903888784350446452241050074, −1.10010248063886839710657736587, 0,
1.10010248063886839710657736587, 2.13903888784350446452241050074, 3.79045627784860642299428481789, 4.95037144105201661968564931705, 5.23101901361279341448717228242, 5.95482100145756852622378928374, 6.97191988730281640323478124432, 7.84357206540786593726080749054, 8.341462190423398317689978413119
