| L(s) = 1 | + 3.01·3-s + 0.418·5-s + 0.933·7-s + 6.07·9-s − 4.47·11-s − 4.34·13-s + 1.25·15-s − 7.93·17-s + 4.27·19-s + 2.81·21-s − 7.70·23-s − 4.82·25-s + 9.27·27-s + 4.27·29-s + 6.35·31-s − 13.4·33-s + 0.390·35-s + 3.56·37-s − 13.0·39-s − 8.23·41-s − 8.01·43-s + 2.54·45-s − 12.1·47-s − 6.12·49-s − 23.8·51-s − 2.39·53-s − 1.87·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.187·5-s + 0.352·7-s + 2.02·9-s − 1.34·11-s − 1.20·13-s + 0.325·15-s − 1.92·17-s + 0.980·19-s + 0.613·21-s − 1.60·23-s − 0.965·25-s + 1.78·27-s + 0.794·29-s + 1.14·31-s − 2.34·33-s + 0.0659·35-s + 0.586·37-s − 2.09·39-s − 1.28·41-s − 1.22·43-s + 0.378·45-s − 1.77·47-s − 0.875·49-s − 3.34·51-s − 0.329·53-s − 0.252·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 3.01T + 3T^{2} \) |
| 5 | \( 1 - 0.418T + 5T^{2} \) |
| 7 | \( 1 - 0.933T + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 13 | \( 1 + 4.34T + 13T^{2} \) |
| 17 | \( 1 + 7.93T + 17T^{2} \) |
| 19 | \( 1 - 4.27T + 19T^{2} \) |
| 23 | \( 1 + 7.70T + 23T^{2} \) |
| 29 | \( 1 - 4.27T + 29T^{2} \) |
| 31 | \( 1 - 6.35T + 31T^{2} \) |
| 37 | \( 1 - 3.56T + 37T^{2} \) |
| 41 | \( 1 + 8.23T + 41T^{2} \) |
| 43 | \( 1 + 8.01T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + 2.39T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 0.530T + 61T^{2} \) |
| 67 | \( 1 + 7.39T + 67T^{2} \) |
| 71 | \( 1 - 7.19T + 71T^{2} \) |
| 73 | \( 1 - 4.19T + 73T^{2} \) |
| 79 | \( 1 - 8.89T + 79T^{2} \) |
| 83 | \( 1 + 0.852T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−7.83349518308451082678263247504, −6.99486332143358628092079820606, −6.31392791362062736013407253260, −5.02102680787612167205554132259, −4.70500817915990998014171454844, −3.75756030136450867537812248663, −2.88880948623094084971360172265, −2.30582466125890490817531864484, −1.79906150441585861341447034519, 0,
1.79906150441585861341447034519, 2.30582466125890490817531864484, 2.88880948623094084971360172265, 3.75756030136450867537812248663, 4.70500817915990998014171454844, 5.02102680787612167205554132259, 6.31392791362062736013407253260, 6.99486332143358628092079820606, 7.83349518308451082678263247504
