L(s) = 1 | − 2.56e3·2-s − 5.90e4·3-s + 4.50e6·4-s + 1.51e8·6-s + 5.58e8·7-s − 6.17e9·8-s + 3.48e9·9-s − 3.70e10·11-s − 2.65e11·12-s − 7.75e10·13-s − 1.43e12·14-s + 6.43e12·16-s + 1.27e13·17-s − 8.95e12·18-s + 2.24e12·19-s − 3.29e13·21-s + 9.52e13·22-s + 2.53e14·23-s + 3.64e14·24-s + 1.99e14·26-s − 2.05e14·27-s + 2.51e15·28-s − 4.09e15·29-s + 2.85e15·31-s − 3.56e15·32-s + 2.19e15·33-s − 3.26e16·34-s + ⋯ |
L(s) = 1 | − 1.77·2-s − 0.577·3-s + 2.14·4-s + 1.02·6-s + 0.747·7-s − 2.03·8-s + 0.333·9-s − 0.431·11-s − 1.23·12-s − 0.156·13-s − 1.32·14-s + 1.46·16-s + 1.53·17-s − 0.591·18-s + 0.0841·19-s − 0.431·21-s + 0.764·22-s + 1.27·23-s + 1.17·24-s + 0.276·26-s − 0.192·27-s + 1.60·28-s − 1.80·29-s + 0.625·31-s − 0.559·32-s + 0.248·33-s − 2.71·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 5.90e4T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 2.56e3T + 2.09e6T^{2} \) |
| 7 | \( 1 - 5.58e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 3.70e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 7.75e10T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.27e13T + 6.90e25T^{2} \) |
| 19 | \( 1 - 2.24e12T + 7.14e26T^{2} \) |
| 23 | \( 1 - 2.53e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 4.09e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 2.85e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 1.07e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 4.19e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 6.32e16T + 2.00e34T^{2} \) |
| 47 | \( 1 + 4.39e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.77e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 2.10e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 4.93e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 5.24e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 4.41e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 2.66e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 9.60e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.56e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 3.75e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 9.05e20T + 5.27e41T^{2} \) |
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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
−10.02872579004032831120655429942, −9.145479024200315161708976956160, −7.900396401174362202995425584036, −7.40509775364316407822202764701, −6.06995715673969741158751596856, −4.92382130144626333009008673417, −3.07458598180431227795612741114, −1.73914552788353847110953939966, −1.01595089010952372099462337105, 0,
1.01595089010952372099462337105, 1.73914552788353847110953939966, 3.07458598180431227795612741114, 4.92382130144626333009008673417, 6.06995715673969741158751596856, 7.40509775364316407822202764701, 7.900396401174362202995425584036, 9.145479024200315161708976956160, 10.02872579004032831120655429942