| L(s) = 1 | + 6·3-s + 5·5-s + 9·9-s + 16·11-s − 58·13-s + 30·15-s + 70·17-s − 4·19-s − 134·23-s + 25·25-s − 108·27-s − 242·29-s − 100·31-s + 96·33-s − 438·37-s − 348·39-s + 138·41-s + 178·43-s + 45·45-s − 22·47-s + 420·51-s + 162·53-s + 80·55-s − 24·57-s + 268·59-s − 250·61-s − 290·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1/3·9-s + 0.438·11-s − 1.23·13-s + 0.516·15-s + 0.998·17-s − 0.0482·19-s − 1.21·23-s + 1/5·25-s − 0.769·27-s − 1.54·29-s − 0.579·31-s + 0.506·33-s − 1.94·37-s − 1.42·39-s + 0.525·41-s + 0.631·43-s + 0.149·45-s − 0.0682·47-s + 1.15·51-s + 0.419·53-s + 0.196·55-s − 0.0557·57-s + 0.591·59-s − 0.524·61-s − 0.553·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 16 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 17 | \( 1 - 70 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 + 134 T + p^{3} T^{2} \) |
| 29 | \( 1 + 242 T + p^{3} T^{2} \) |
| 31 | \( 1 + 100 T + p^{3} T^{2} \) |
| 37 | \( 1 + 438 T + p^{3} T^{2} \) |
| 41 | \( 1 - 138 T + p^{3} T^{2} \) |
| 43 | \( 1 - 178 T + p^{3} T^{2} \) |
| 47 | \( 1 + 22 T + p^{3} T^{2} \) |
| 53 | \( 1 - 162 T + p^{3} T^{2} \) |
| 59 | \( 1 - 268 T + p^{3} T^{2} \) |
| 61 | \( 1 + 250 T + p^{3} T^{2} \) |
| 67 | \( 1 - 422 T + p^{3} T^{2} \) |
| 71 | \( 1 + 12 p T + p^{3} T^{2} \) |
| 73 | \( 1 + 306 T + p^{3} T^{2} \) |
| 79 | \( 1 + 456 T + p^{3} T^{2} \) |
| 83 | \( 1 + 434 T + p^{3} T^{2} \) |
| 89 | \( 1 - 726 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1378 T + p^{3} T^{2} \) |
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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
−8.531652953680097538273352115428, −7.61040323647552642888281353468, −7.19980034359261653906805193602, −5.93984654727745867987529720821, −5.28603194217139024704225089007, −4.06994420605845543336207782654, −3.33132203974329085362395005625, −2.35571767943405065688381189042, −1.64530064310203523071897121100, 0,
1.64530064310203523071897121100, 2.35571767943405065688381189042, 3.33132203974329085362395005625, 4.06994420605845543336207782654, 5.28603194217139024704225089007, 5.93984654727745867987529720821, 7.19980034359261653906805193602, 7.61040323647552642888281353468, 8.531652953680097538273352115428
