| L(s) = 1 | − 4·3-s + 16·7-s − 11·9-s + 36·11-s − 42·13-s + 110·17-s − 116·19-s − 64·21-s + 16·23-s + 152·27-s − 198·29-s − 240·31-s − 144·33-s − 258·37-s + 168·39-s + 442·41-s + 292·43-s + 392·47-s − 87·49-s − 440·51-s + 142·53-s + 464·57-s − 348·59-s + 570·61-s − 176·63-s − 692·67-s − 64·69-s + ⋯ |
| L(s) = 1 | − 0.769·3-s + 0.863·7-s − 0.407·9-s + 0.986·11-s − 0.896·13-s + 1.56·17-s − 1.40·19-s − 0.665·21-s + 0.145·23-s + 1.08·27-s − 1.26·29-s − 1.39·31-s − 0.759·33-s − 1.14·37-s + 0.689·39-s + 1.68·41-s + 1.03·43-s + 1.21·47-s − 0.253·49-s − 1.20·51-s + 0.368·53-s + 1.07·57-s − 0.767·59-s + 1.19·61-s − 0.351·63-s − 1.26·67-s − 0.111·69-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 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 42 T + p^{3} T^{2} \) |
| 17 | \( 1 - 110 T + p^{3} T^{2} \) |
| 19 | \( 1 + 116 T + p^{3} T^{2} \) |
| 23 | \( 1 - 16 T + p^{3} T^{2} \) |
| 29 | \( 1 + 198 T + p^{3} T^{2} \) |
| 31 | \( 1 + 240 T + p^{3} T^{2} \) |
| 37 | \( 1 + 258 T + p^{3} T^{2} \) |
| 41 | \( 1 - 442 T + p^{3} T^{2} \) |
| 43 | \( 1 - 292 T + p^{3} T^{2} \) |
| 47 | \( 1 - 392 T + p^{3} T^{2} \) |
| 53 | \( 1 - 142 T + p^{3} T^{2} \) |
| 59 | \( 1 + 348 T + p^{3} T^{2} \) |
| 61 | \( 1 - 570 T + p^{3} T^{2} \) |
| 67 | \( 1 + 692 T + p^{3} T^{2} \) |
| 71 | \( 1 + 168 T + p^{3} T^{2} \) |
| 73 | \( 1 - 134 T + p^{3} T^{2} \) |
| 79 | \( 1 + 784 T + p^{3} T^{2} \) |
| 83 | \( 1 + 564 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1034 T + p^{3} T^{2} \) |
| 97 | \( 1 - 382 T + p^{3} T^{2} \) |
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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
−8.741319773006440203323094931671, −7.70830522624835213176117818082, −7.11346041672006348402113594109, −5.95491748426959436023249829427, −5.50758149764219523481945357671, −4.55930156556841928628265707567, −3.66196236461227128796310564937, −2.30806526050957932491688580605, −1.22609702552539657233266644505, 0,
1.22609702552539657233266644505, 2.30806526050957932491688580605, 3.66196236461227128796310564937, 4.55930156556841928628265707567, 5.50758149764219523481945357671, 5.95491748426959436023249829427, 7.11346041672006348402113594109, 7.70830522624835213176117818082, 8.741319773006440203323094931671
