| L(s) = 1 | + 3.60·3-s + 2.21·5-s − 4.42·7-s − 14.0·9-s + 12.3·11-s − 48.6·13-s + 7.97·15-s − 101.·17-s + 59.2·19-s − 15.9·21-s − 18.0·23-s − 120.·25-s − 147.·27-s − 29·29-s − 20.8·31-s + 44.6·33-s − 9.77·35-s + 101.·37-s − 175.·39-s + 40.7·41-s − 152.·43-s − 30.9·45-s − 121.·47-s − 323.·49-s − 365.·51-s + 177.·53-s + 27.4·55-s + ⋯ |
| L(s) = 1 | + 0.693·3-s + 0.197·5-s − 0.238·7-s − 0.518·9-s + 0.339·11-s − 1.03·13-s + 0.137·15-s − 1.44·17-s + 0.715·19-s − 0.165·21-s − 0.163·23-s − 0.960·25-s − 1.05·27-s − 0.185·29-s − 0.120·31-s + 0.235·33-s − 0.0472·35-s + 0.449·37-s − 0.719·39-s + 0.155·41-s − 0.540·43-s − 0.102·45-s − 0.376·47-s − 0.942·49-s − 1.00·51-s + 0.460·53-s + 0.0671·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 - 3.60T + 27T^{2} \) |
| 5 | \( 1 - 2.21T + 125T^{2} \) |
| 7 | \( 1 + 4.42T + 343T^{2} \) |
| 11 | \( 1 - 12.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 48.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 59.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 18.0T + 1.21e4T^{2} \) |
| 31 | \( 1 + 20.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 101.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 40.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 152.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 121.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 177.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 109.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 61.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 471.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 546.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 169.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 184.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 210.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.39e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.09e3T + 9.12e5T^{2} \) |
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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
−9.890985291080870011127910076927, −9.333012596095464937501338639715, −8.442331954645941093147449333367, −7.50997477847901691637331863397, −6.51176370568029203083249794954, −5.41021900236051326794866398169, −4.19549282936872331053748703961, −2.98055685404707266727496648212, −1.99346490825132851247852964442, 0,
1.99346490825132851247852964442, 2.98055685404707266727496648212, 4.19549282936872331053748703961, 5.41021900236051326794866398169, 6.51176370568029203083249794954, 7.50997477847901691637331863397, 8.442331954645941093147449333367, 9.333012596095464937501338639715, 9.890985291080870011127910076927
