| L(s) = 1 | + 2-s + 1.67·3-s + 4-s − 0.566·5-s + 1.67·6-s + 3.40·7-s + 8-s − 0.199·9-s − 0.566·10-s + 1.67·12-s + 1.35·13-s + 3.40·14-s − 0.948·15-s + 16-s + 4.30·17-s − 0.199·18-s − 19-s − 0.566·20-s + 5.69·21-s − 2.71·23-s + 1.67·24-s − 4.67·25-s + 1.35·26-s − 5.35·27-s + 3.40·28-s + 5.64·29-s − 0.948·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.966·3-s + 0.5·4-s − 0.253·5-s + 0.683·6-s + 1.28·7-s + 0.353·8-s − 0.0664·9-s − 0.179·10-s + 0.483·12-s + 0.375·13-s + 0.908·14-s − 0.244·15-s + 0.250·16-s + 1.04·17-s − 0.0469·18-s − 0.229·19-s − 0.126·20-s + 1.24·21-s − 0.565·23-s + 0.341·24-s − 0.935·25-s + 0.265·26-s − 1.03·27-s + 0.642·28-s + 1.04·29-s − 0.173·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.046718211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.046718211\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.67T + 3T^{2} \) |
| 5 | \( 1 + 0.566T + 5T^{2} \) |
| 7 | \( 1 - 3.40T + 7T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 - 4.30T + 17T^{2} \) |
| 23 | \( 1 + 2.71T + 23T^{2} \) |
| 29 | \( 1 - 5.64T + 29T^{2} \) |
| 31 | \( 1 - 9.62T + 31T^{2} \) |
| 37 | \( 1 - 3.63T + 37T^{2} \) |
| 41 | \( 1 + 3.64T + 41T^{2} \) |
| 43 | \( 1 + 7.94T + 43T^{2} \) |
| 47 | \( 1 - 13.5T + 47T^{2} \) |
| 53 | \( 1 - 4.25T + 53T^{2} \) |
| 59 | \( 1 - 7.97T + 59T^{2} \) |
| 61 | \( 1 - 9.37T + 61T^{2} \) |
| 67 | \( 1 - 1.01T + 67T^{2} \) |
| 71 | \( 1 - 6.30T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 7.11T + 79T^{2} \) |
| 83 | \( 1 + 1.02T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 - 7.04T + 97T^{2} \) |
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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
−8.279877031665342665604170372821, −7.80322297065595968216663254420, −6.96619875614823865872347626348, −5.95356968097107827814997020372, −5.33975774143284671681356588616, −4.40160810722695009920330954667, −3.85051205161488952541515971454, −2.91472850206919615310001540957, −2.19230459402056782753035371882, −1.16994541459959863591064273553,
1.16994541459959863591064273553, 2.19230459402056782753035371882, 2.91472850206919615310001540957, 3.85051205161488952541515971454, 4.40160810722695009920330954667, 5.33975774143284671681356588616, 5.95356968097107827814997020372, 6.96619875614823865872347626348, 7.80322297065595968216663254420, 8.279877031665342665604170372821
