| L(s) = 1 | − 4·3-s − 16·7-s − 11·9-s − 60·11-s + 86·13-s − 18·17-s + 44·19-s + 64·21-s + 48·23-s + 152·27-s + 186·29-s − 176·31-s + 240·33-s + 254·37-s − 344·39-s + 186·41-s + 100·43-s + 168·47-s − 87·49-s + 72·51-s − 498·53-s − 176·57-s − 252·59-s + 58·61-s + 176·63-s + 1.03e3·67-s − 192·69-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 0.863·7-s − 0.407·9-s − 1.64·11-s + 1.83·13-s − 0.256·17-s + 0.531·19-s + 0.665·21-s + 0.435·23-s + 1.08·27-s + 1.19·29-s − 1.01·31-s + 1.26·33-s + 1.12·37-s − 1.41·39-s + 0.708·41-s + 0.354·43-s + 0.521·47-s − 0.253·49-s + 0.197·51-s − 1.29·53-s − 0.408·57-s − 0.556·59-s + 0.121·61-s + 0.351·63-s + 1.88·67-s − 0.334·69-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 | \( 1 + 60 T + p^{3} T^{2} \) |
| 13 | \( 1 - 86 T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 - 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 - 186 T + p^{3} T^{2} \) |
| 31 | \( 1 + 176 T + p^{3} T^{2} \) |
| 37 | \( 1 - 254 T + p^{3} T^{2} \) |
| 41 | \( 1 - 186 T + p^{3} T^{2} \) |
| 43 | \( 1 - 100 T + p^{3} T^{2} \) |
| 47 | \( 1 - 168 T + p^{3} T^{2} \) |
| 53 | \( 1 + 498 T + p^{3} T^{2} \) |
| 59 | \( 1 + 252 T + p^{3} T^{2} \) |
| 61 | \( 1 - 58 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1036 T + p^{3} T^{2} \) |
| 71 | \( 1 + 168 T + p^{3} T^{2} \) |
| 73 | \( 1 + 506 T + p^{3} T^{2} \) |
| 79 | \( 1 + 272 T + p^{3} T^{2} \) |
| 83 | \( 1 + 948 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1014 T + p^{3} T^{2} \) |
| 97 | \( 1 - 766 T + p^{3} T^{2} \) |
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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
−8.604200784338557105471300236436, −7.912622166930640061152208812932, −6.85819806008588924965344057091, −6.02677138651078518313750108629, −5.60121588368688315512263015304, −4.59731273365971337735128944840, −3.37193431047213631867845280852, −2.65119767680412610029524300173, −1.01292286925120817477331675289, 0,
1.01292286925120817477331675289, 2.65119767680412610029524300173, 3.37193431047213631867845280852, 4.59731273365971337735128944840, 5.60121588368688315512263015304, 6.02677138651078518313750108629, 6.85819806008588924965344057091, 7.912622166930640061152208812932, 8.604200784338557105471300236436
