| 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
| \(L(2)\) |
\(\approx\) |
\(3.333950470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.333950470\) |
| \(L(\frac{5}{2})\) |
|
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
−9.105921820664199125260757361906, −8.300637610774630132117018879082, −7.37795196084154701844332503743, −7.14668462352730111396516721517, −5.75502762664416276467004496662, −4.76882678009111725199752431059, −3.57393278607174061446424571583, −2.85469636941045360769011860044, −2.48468896344146973630823661749, −0.78880587193246888501100179150,
0.78880587193246888501100179150, 2.48468896344146973630823661749, 2.85469636941045360769011860044, 3.57393278607174061446424571583, 4.76882678009111725199752431059, 5.75502762664416276467004496662, 7.14668462352730111396516721517, 7.37795196084154701844332503743, 8.300637610774630132117018879082, 9.105921820664199125260757361906
