| L(s) = 1 | + 4·5-s − 3·7-s − 28·11-s − 11·13-s + 44·17-s − 29·19-s − 172·23-s − 109·25-s + 192·29-s − 116·31-s − 12·35-s − 69·37-s + 384·41-s − 328·43-s − 156·47-s − 334·49-s − 392·53-s − 112·55-s − 412·59-s − 425·61-s − 44·65-s − 257·67-s + 1.00e3·71-s − 359·73-s + 84·77-s − 877·79-s + 328·83-s + ⋯ |
| L(s) = 1 | + 0.357·5-s − 0.161·7-s − 0.767·11-s − 0.234·13-s + 0.627·17-s − 0.350·19-s − 1.55·23-s − 0.871·25-s + 1.22·29-s − 0.672·31-s − 0.0579·35-s − 0.306·37-s + 1.46·41-s − 1.16·43-s − 0.484·47-s − 0.973·49-s − 1.01·53-s − 0.274·55-s − 0.909·59-s − 0.892·61-s − 0.0839·65-s − 0.468·67-s + 1.67·71-s − 0.575·73-s + 0.124·77-s − 1.24·79-s + 0.433·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + 3 T + p^{3} T^{2} \) |
| 11 | \( 1 + 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 11 T + p^{3} T^{2} \) |
| 17 | \( 1 - 44 T + p^{3} T^{2} \) |
| 19 | \( 1 + 29 T + p^{3} T^{2} \) |
| 23 | \( 1 + 172 T + p^{3} T^{2} \) |
| 29 | \( 1 - 192 T + p^{3} T^{2} \) |
| 31 | \( 1 + 116 T + p^{3} T^{2} \) |
| 37 | \( 1 + 69 T + p^{3} T^{2} \) |
| 41 | \( 1 - 384 T + p^{3} T^{2} \) |
| 43 | \( 1 + 328 T + p^{3} T^{2} \) |
| 47 | \( 1 + 156 T + p^{3} T^{2} \) |
| 53 | \( 1 + 392 T + p^{3} T^{2} \) |
| 59 | \( 1 + 412 T + p^{3} T^{2} \) |
| 61 | \( 1 + 425 T + p^{3} T^{2} \) |
| 67 | \( 1 + 257 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1000 T + p^{3} T^{2} \) |
| 73 | \( 1 + 359 T + p^{3} T^{2} \) |
| 79 | \( 1 + 877 T + p^{3} T^{2} \) |
| 83 | \( 1 - 328 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1572 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1483 T + p^{3} T^{2} \) |
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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
−10.13615875415260965041301235033, −9.618297746063117373476691161073, −8.330233698610467692734342720940, −7.64007195694450370292373437341, −6.38794709280120521242519642887, −5.56279361003014826347240028963, −4.41035969817330769825249527008, −3.07638765987746695352789603058, −1.82088973550670102006483342519, 0,
1.82088973550670102006483342519, 3.07638765987746695352789603058, 4.41035969817330769825249527008, 5.56279361003014826347240028963, 6.38794709280120521242519642887, 7.64007195694450370292373437341, 8.330233698610467692734342720940, 9.618297746063117373476691161073, 10.13615875415260965041301235033
