| L(s) = 1 | + 1.50·2-s + 0.267·4-s + 4.11·5-s + 1.26·7-s − 2.60·8-s + 6.19·10-s + 2.60·11-s − 3.46·13-s + 1.90·14-s − 4.46·16-s − 19-s + 1.10·20-s + 3.92·22-s − 1.10·23-s + 11.9·25-s − 5.21·26-s + 0.339·28-s − 8.63·29-s + 10.6·31-s − 1.50·32-s + 5.21·35-s + 4.19·37-s − 1.50·38-s − 10.7·40-s − 3.71·41-s − 2.73·43-s + 0.698·44-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.133·4-s + 1.84·5-s + 0.479·7-s − 0.922·8-s + 1.95·10-s + 0.786·11-s − 0.960·13-s + 0.510·14-s − 1.11·16-s − 0.229·19-s + 0.246·20-s + 0.837·22-s − 0.229·23-s + 2.38·25-s − 1.02·26-s + 0.0642·28-s − 1.60·29-s + 1.91·31-s − 0.266·32-s + 0.881·35-s + 0.689·37-s − 0.244·38-s − 1.69·40-s − 0.579·41-s − 0.416·43-s + 0.105·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.901981088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.901981088\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.50T + 2T^{2} \) |
| 5 | \( 1 - 4.11T + 5T^{2} \) |
| 7 | \( 1 - 1.26T + 7T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 1.10T + 23T^{2} \) |
| 29 | \( 1 + 8.63T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 - 4.19T + 37T^{2} \) |
| 41 | \( 1 + 3.71T + 41T^{2} \) |
| 43 | \( 1 + 2.73T + 43T^{2} \) |
| 47 | \( 1 + 4.51T + 47T^{2} \) |
| 53 | \( 1 + 4.51T + 53T^{2} \) |
| 59 | \( 1 + 7.12T + 59T^{2} \) |
| 61 | \( 1 + 7.19T + 61T^{2} \) |
| 67 | \( 1 + 5.73T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 + 9T + 79T^{2} \) |
| 83 | \( 1 + 7.52T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 + 7.66T + 97T^{2} \) |
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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
−11.01423526293359160836111144970, −9.703378047061319108982288685914, −9.509909474097867304340314943611, −8.311545318791799061823375592705, −6.75794540506589780098121444650, −6.05892329293598539375529207567, −5.21022895178882818642038729400, −4.44254791916248472279817020057, −2.93250915065198606474738399801, −1.79249472125450849546359112909,
1.79249472125450849546359112909, 2.93250915065198606474738399801, 4.44254791916248472279817020057, 5.21022895178882818642038729400, 6.05892329293598539375529207567, 6.75794540506589780098121444650, 8.311545318791799061823375592705, 9.509909474097867304340314943611, 9.703378047061319108982288685914, 11.01423526293359160836111144970
