| L(s) = 1 | − 2·5-s + 4.79·7-s + 11-s − 13-s + 4·17-s + 5.79·19-s − 4.79·23-s − 25-s − 5.58·29-s − 9.58·35-s + 4·37-s + 2.79·41-s + 11.1·43-s + 5.58·47-s + 15.9·49-s + 9.37·53-s − 2·55-s − 3.58·59-s + 11.5·61-s + 2·65-s − 7.16·67-s − 6.20·73-s + 4.79·77-s − 14.7·79-s − 14.3·83-s − 8·85-s − 0.417·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.81·7-s + 0.301·11-s − 0.277·13-s + 0.970·17-s + 1.32·19-s − 0.999·23-s − 0.200·25-s − 1.03·29-s − 1.61·35-s + 0.657·37-s + 0.435·41-s + 1.70·43-s + 0.814·47-s + 2.27·49-s + 1.28·53-s − 0.269·55-s − 0.466·59-s + 1.48·61-s + 0.248·65-s − 0.875·67-s − 0.726·73-s + 0.546·77-s − 1.65·79-s − 1.57·83-s − 0.867·85-s − 0.0442·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.218469336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.218469336\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 4.79T + 7T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 5.79T + 19T^{2} \) |
| 23 | \( 1 + 4.79T + 23T^{2} \) |
| 29 | \( 1 + 5.58T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 2.79T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 5.58T + 47T^{2} \) |
| 53 | \( 1 - 9.37T + 53T^{2} \) |
| 59 | \( 1 + 3.58T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 7.16T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 6.20T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 + 0.417T + 89T^{2} \) |
| 97 | \( 1 + 8.74T + 97T^{2} \) |
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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
−7.988433916500393192326368670585, −7.60779276745422617827574217785, −7.19847008284790552790441598611, −5.65112182419634948296619279151, −5.50364143120136031488799463813, −4.24567551160039909062875862356, −4.07617823187376759704315766619, −2.84652135840024851326499181661, −1.76940162919110718232451480088, −0.860193658470505192894362353021,
0.860193658470505192894362353021, 1.76940162919110718232451480088, 2.84652135840024851326499181661, 4.07617823187376759704315766619, 4.24567551160039909062875862356, 5.50364143120136031488799463813, 5.65112182419634948296619279151, 7.19847008284790552790441598611, 7.60779276745422617827574217785, 7.988433916500393192326368670585
