| L(s) = 1 | + 4.39·2-s − 3·3-s + 11.3·4-s + 2.45·5-s − 13.1·6-s + 13.1·7-s + 14.6·8-s + 9·9-s + 10.7·10-s − 11·11-s − 33.9·12-s + 12.0·13-s + 57.8·14-s − 7.35·15-s − 26.2·16-s − 60.4·17-s + 39.5·18-s + 78.9·19-s + 27.7·20-s − 39.4·21-s − 48.3·22-s − 209.·23-s − 43.9·24-s − 118.·25-s + 52.9·26-s − 27·27-s + 149.·28-s + ⋯ |
| L(s) = 1 | + 1.55·2-s − 0.577·3-s + 1.41·4-s + 0.219·5-s − 0.897·6-s + 0.710·7-s + 0.647·8-s + 0.333·9-s + 0.340·10-s − 0.301·11-s − 0.817·12-s + 0.256·13-s + 1.10·14-s − 0.126·15-s − 0.409·16-s − 0.862·17-s + 0.518·18-s + 0.953·19-s + 0.310·20-s − 0.410·21-s − 0.468·22-s − 1.89·23-s − 0.373·24-s − 0.951·25-s + 0.399·26-s − 0.192·27-s + 1.00·28-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 - 4.39T + 8T^{2} \) |
| 5 | \( 1 - 2.45T + 125T^{2} \) |
| 7 | \( 1 - 13.1T + 343T^{2} \) |
| 13 | \( 1 - 12.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 60.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 78.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 209.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 276.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 89.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 201.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 341.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 540.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 50.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 526.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 466.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 661.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 597.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 431.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 358.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 931.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.48e3T + 9.12e5T^{2} \) |
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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
−8.137254581998217015096555790411, −7.40961307070312696650863970835, −6.43328261042008827556872532644, −5.74978814947371357929458647190, −5.26592887724435718105879322678, −4.30773485685555032575008981759, −3.79099269868175101546454698539, −2.48921792020245972399596775946, −1.65968035853535469564762473282, 0,
1.65968035853535469564762473282, 2.48921792020245972399596775946, 3.79099269868175101546454698539, 4.30773485685555032575008981759, 5.26592887724435718105879322678, 5.74978814947371357929458647190, 6.43328261042008827556872532644, 7.40961307070312696650863970835, 8.137254581998217015096555790411
