| L(s) = 1 | + 8.47·3-s − 0.885·5-s − 3.81·7-s + 44.8·9-s − 52.3·11-s + 8.06·13-s − 7.50·15-s − 17·17-s − 66.5·19-s − 32.3·21-s − 180.·23-s − 124.·25-s + 151.·27-s + 41.2·29-s + 34.9·31-s − 443.·33-s + 3.38·35-s − 130.·37-s + 68.3·39-s − 17.9·41-s + 277.·43-s − 39.7·45-s − 463.·47-s − 328.·49-s − 144.·51-s + 329.·53-s + 46.3·55-s + ⋯ |
| L(s) = 1 | + 1.63·3-s − 0.0792·5-s − 0.206·7-s + 1.66·9-s − 1.43·11-s + 0.171·13-s − 0.129·15-s − 0.242·17-s − 0.803·19-s − 0.336·21-s − 1.63·23-s − 0.993·25-s + 1.07·27-s + 0.264·29-s + 0.202·31-s − 2.34·33-s + 0.0163·35-s − 0.579·37-s + 0.280·39-s − 0.0682·41-s + 0.984·43-s − 0.131·45-s − 1.43·47-s − 0.957·49-s − 0.395·51-s + 0.855·53-s + 0.113·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 3 | \( 1 - 8.47T + 27T^{2} \) |
| 5 | \( 1 + 0.885T + 125T^{2} \) |
| 7 | \( 1 + 3.81T + 343T^{2} \) |
| 11 | \( 1 + 52.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8.06T + 2.19e3T^{2} \) |
| 19 | \( 1 + 66.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 180.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 41.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 34.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 130.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 17.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 277.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 463.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 329.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 678.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 340.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 15.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 670.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 193.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 865.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 379.T + 9.12e5T^{2} \) |
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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
−8.934063486908401225770246818747, −8.106491973262484663525216324316, −7.87283738894892906112458329569, −6.75590919683557910853113929652, −5.64992965366048427980939959113, −4.40044688519933245180067936841, −3.58528683959378001714736823639, −2.58953477684582338359863266976, −1.90803195389622421785264383002, 0,
1.90803195389622421785264383002, 2.58953477684582338359863266976, 3.58528683959378001714736823639, 4.40044688519933245180067936841, 5.64992965366048427980939959113, 6.75590919683557910853113929652, 7.87283738894892906112458329569, 8.106491973262484663525216324316, 8.934063486908401225770246818747
