| 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
−8.565117193839080018402111056529, −7.76045880219095782217809542990, −6.74137077031353385246895705067, −6.21839775245342571090445578059, −5.38344254778248489216121278340, −4.47733196265797379518440683215, −4.09098420430934023309013479006, −1.99320267195411194527290335547, −1.21352588292291323255874340499, 0,
1.21352588292291323255874340499, 1.99320267195411194527290335547, 4.09098420430934023309013479006, 4.47733196265797379518440683215, 5.38344254778248489216121278340, 6.21839775245342571090445578059, 6.74137077031353385246895705067, 7.76045880219095782217809542990, 8.565117193839080018402111056529
