| L(s) = 1 | − 350.·2-s + 5.58e4·3-s − 4.01e5·4-s + 2.41e6·5-s − 1.96e7·6-s + 9.44e7·7-s + 3.24e8·8-s + 1.95e9·9-s − 8.48e8·10-s + 1.32e10·11-s − 2.24e10·12-s + 7.42e10·13-s − 3.31e10·14-s + 1.35e11·15-s + 9.63e10·16-s + 8.03e11·17-s − 6.86e11·18-s + 8.34e11·19-s − 9.70e11·20-s + 5.27e12·21-s − 4.63e12·22-s − 1.72e13·23-s + 1.81e13·24-s − 1.32e13·25-s − 2.60e13·26-s + 4.44e13·27-s − 3.78e13·28-s + ⋯ |
| L(s) = 1 | − 0.484·2-s + 1.63·3-s − 0.765·4-s + 0.553·5-s − 0.794·6-s + 0.884·7-s + 0.855·8-s + 1.68·9-s − 0.268·10-s + 1.69·11-s − 1.25·12-s + 1.94·13-s − 0.428·14-s + 0.907·15-s + 0.350·16-s + 1.64·17-s − 0.816·18-s + 0.593·19-s − 0.423·20-s + 1.44·21-s − 0.819·22-s − 1.99·23-s + 1.40·24-s − 0.693·25-s − 0.940·26-s + 1.12·27-s − 0.676·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(4.605972067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.605972067\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 47 | \( 1 + 1.11e15T \) |
| good | 2 | \( 1 + 350.T + 5.24e5T^{2} \) |
| 3 | \( 1 - 5.58e4T + 1.16e9T^{2} \) |
| 5 | \( 1 - 2.41e6T + 1.90e13T^{2} \) |
| 7 | \( 1 - 9.44e7T + 1.13e16T^{2} \) |
| 11 | \( 1 - 1.32e10T + 6.11e19T^{2} \) |
| 13 | \( 1 - 7.42e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 8.03e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 8.34e11T + 1.97e24T^{2} \) |
| 23 | \( 1 + 1.72e13T + 7.46e25T^{2} \) |
| 29 | \( 1 + 3.50e13T + 6.10e27T^{2} \) |
| 31 | \( 1 - 1.81e14T + 2.16e28T^{2} \) |
| 37 | \( 1 + 6.44e13T + 6.24e29T^{2} \) |
| 41 | \( 1 + 5.90e14T + 4.39e30T^{2} \) |
| 43 | \( 1 + 4.36e14T + 1.08e31T^{2} \) |
| 53 | \( 1 + 3.01e15T + 5.77e32T^{2} \) |
| 59 | \( 1 + 8.65e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 6.02e16T + 8.34e33T^{2} \) |
| 67 | \( 1 + 1.13e17T + 4.95e34T^{2} \) |
| 71 | \( 1 + 2.40e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 6.22e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 9.82e16T + 1.13e36T^{2} \) |
| 83 | \( 1 + 9.34e17T + 2.90e36T^{2} \) |
| 89 | \( 1 - 2.59e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 8.91e18T + 5.60e37T^{2} \) |
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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
−11.84051795642244270094780777588, −10.09045738233944057223097892748, −9.323850000470096193077840910910, −8.410696834474234390667360515413, −7.82926715103631828950482976487, −5.95656712348986348806563917678, −4.14136590088747998376673137611, −3.48772657501081223692596429608, −1.55855264140461672202564891673, −1.32465840224683399174934946372,
1.32465840224683399174934946372, 1.55855264140461672202564891673, 3.48772657501081223692596429608, 4.14136590088747998376673137611, 5.95656712348986348806563917678, 7.82926715103631828950482976487, 8.410696834474234390667360515413, 9.323850000470096193077840910910, 10.09045738233944057223097892748, 11.84051795642244270094780777588
