| L(s) = 1 | + 1.12e3·3-s + 1.56e4·5-s − 3.24e5·7-s − 3.26e5·9-s + 1.64e6·11-s + 6.26e6·13-s + 1.75e7·15-s + 1.66e8·17-s − 3.12e8·19-s − 3.65e8·21-s + 6.32e8·23-s + 2.44e8·25-s − 2.16e9·27-s − 2.82e9·29-s − 7.61e9·31-s + 1.85e9·33-s − 5.07e9·35-s + 1.99e10·37-s + 7.05e9·39-s − 4.69e10·41-s + 7.85e9·43-s − 5.10e9·45-s − 8.31e10·47-s + 8.39e9·49-s + 1.87e11·51-s − 1.19e11·53-s + 2.57e10·55-s + ⋯ |
| L(s) = 1 | + 0.891·3-s + 0.447·5-s − 1.04·7-s − 0.205·9-s + 0.280·11-s + 0.360·13-s + 0.398·15-s + 1.67·17-s − 1.52·19-s − 0.929·21-s + 0.890·23-s + 0.199·25-s − 1.07·27-s − 0.882·29-s − 1.54·31-s + 0.249·33-s − 0.466·35-s + 1.27·37-s + 0.321·39-s − 1.54·41-s + 0.189·43-s − 0.0916·45-s − 1.12·47-s + 0.0866·49-s + 1.49·51-s − 0.739·53-s + 0.125·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 1.56e4T \) |
| good | 3 | \( 1 - 1.12e3T + 1.59e6T^{2} \) |
| 7 | \( 1 + 3.24e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 1.64e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 6.26e6T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.66e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 3.12e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 6.32e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 2.82e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 7.61e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.99e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 4.69e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 7.85e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + 8.31e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.19e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 4.20e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 4.15e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 1.02e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 4.00e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 5.55e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 1.60e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 2.64e11T + 8.87e24T^{2} \) |
| 89 | \( 1 + 3.69e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.00e13T + 6.73e25T^{2} \) |
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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
−11.09766326486535470085612114012, −9.820858258352169516417944735430, −9.089045204781870310743987576336, −7.996473788924319002631776168333, −6.63051268245743834167687415439, −5.55290673607762340189600260628, −3.72469432397198523262958644742, −2.91898109674791511098151481432, −1.61753497332193712697640038787, 0,
1.61753497332193712697640038787, 2.91898109674791511098151481432, 3.72469432397198523262958644742, 5.55290673607762340189600260628, 6.63051268245743834167687415439, 7.996473788924319002631776168333, 9.089045204781870310743987576336, 9.820858258352169516417944735430, 11.09766326486535470085612114012
