| L(s) = 1 | − 5·5-s + 16.4·7-s − 57.6·11-s + 38.7·13-s − 31.4·17-s + 70.6·19-s + 7.37·23-s + 25·25-s + 17.1·29-s − 50.9·31-s − 82.0·35-s − 159.·37-s + 63.2·41-s − 84.6·43-s − 434.·47-s − 73.6·49-s − 138.·53-s + 288.·55-s + 631.·59-s + 82.8·61-s − 193.·65-s − 431.·67-s − 450.·71-s − 33.5·73-s − 945.·77-s + 509.·79-s + 811.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.886·7-s − 1.57·11-s + 0.827·13-s − 0.448·17-s + 0.852·19-s + 0.0668·23-s + 0.200·25-s + 0.109·29-s − 0.295·31-s − 0.396·35-s − 0.709·37-s + 0.240·41-s − 0.300·43-s − 1.34·47-s − 0.214·49-s − 0.358·53-s + 0.706·55-s + 1.39·59-s + 0.173·61-s − 0.370·65-s − 0.787·67-s − 0.752·71-s − 0.0538·73-s − 1.39·77-s + 0.725·79-s + 1.07·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 - 16.4T + 343T^{2} \) |
| 11 | \( 1 + 57.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 31.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 70.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 7.37T + 1.21e4T^{2} \) |
| 29 | \( 1 - 17.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 50.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 159.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 63.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 84.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 434.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 138.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 631.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 82.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 431.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 450.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 33.5T + 3.89e5T^{2} \) |
| 79 | \( 1 - 509.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 811.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 499.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 625.T + 9.12e5T^{2} \) |
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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
−8.920997719762445011616983049624, −8.123223048880041412392492650052, −7.66883830973207514300053793824, −6.62993660017870673966316052575, −5.42249356570447592576916578410, −4.87687871983544592852167946789, −3.72269555153788799733199914875, −2.65916665768805068549757905938, −1.41329647326403708474829547054, 0,
1.41329647326403708474829547054, 2.65916665768805068549757905938, 3.72269555153788799733199914875, 4.87687871983544592852167946789, 5.42249356570447592576916578410, 6.62993660017870673966316052575, 7.66883830973207514300053793824, 8.123223048880041412392492650052, 8.920997719762445011616983049624
