| L(s) = 1 | + 3.27·2-s + 3·3-s + 2.72·4-s + 1.50·5-s + 9.82·6-s + 19.6·7-s − 17.2·8-s + 9·9-s + 4.93·10-s + 11·11-s + 8.16·12-s + 56.4·13-s + 64.4·14-s + 4.51·15-s − 78.3·16-s + 136.·17-s + 29.4·18-s + 90.4·19-s + 4.10·20-s + 59.0·21-s + 36.0·22-s + 28.1·23-s − 51.8·24-s − 122.·25-s + 184.·26-s + 27·27-s + 53.5·28-s + ⋯ |
| L(s) = 1 | + 1.15·2-s + 0.577·3-s + 0.340·4-s + 0.134·5-s + 0.668·6-s + 1.06·7-s − 0.763·8-s + 0.333·9-s + 0.156·10-s + 0.301·11-s + 0.196·12-s + 1.20·13-s + 1.23·14-s + 0.0778·15-s − 1.22·16-s + 1.95·17-s + 0.385·18-s + 1.09·19-s + 0.0458·20-s + 0.613·21-s + 0.349·22-s + 0.254·23-s − 0.440·24-s − 0.981·25-s + 1.39·26-s + 0.192·27-s + 0.361·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.571857947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.571857947\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 - 3.27T + 8T^{2} \) |
| 5 | \( 1 - 1.50T + 125T^{2} \) |
| 7 | \( 1 - 19.6T + 343T^{2} \) |
| 13 | \( 1 - 56.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 136.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 90.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 28.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 275.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 15.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 205.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 58.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 328.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 119.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 493.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 579.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 359.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 312.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 221.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 278.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 618.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.62e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3T + 9.12e5T^{2} \) |
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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
−8.835543812727304522244592032688, −7.85905130802347728970007000052, −7.37353493702253343001334021518, −5.98944625365727579960804866672, −5.56623304146170237768937112555, −4.73140532024648789630226241600, −3.63197419432140457159879390957, −3.41261437092468906616137184589, −1.98958611052674235697675687613, −1.04707869326751348062771763005,
1.04707869326751348062771763005, 1.98958611052674235697675687613, 3.41261437092468906616137184589, 3.63197419432140457159879390957, 4.73140532024648789630226241600, 5.56623304146170237768937112555, 5.98944625365727579960804866672, 7.37353493702253343001334021518, 7.85905130802347728970007000052, 8.835543812727304522244592032688
